What are the Jones polynomials for the torus links and the closure of the other braid word below?












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I am working on a project to determine the Jones polynomial for the torus links and a class of links which I call tst links. Their braid words are respectively given by
$$(sigma_1 sigma_2 cdots sigma_{p-1})^q in B_p$$
and
$$(sigma_1^2 sigma_2 sigma_3 cdots sigma_p)^q in B_{p+1}$$



I would like to know if anyone familiar with the Ocneanu approach, Chern-Simons Theory approach, and the Burau representations approach can evaluate the Jones polynomials and give a clear, detailed explanation.










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    This is to extend the question math.stackexchange.com/questions/235246/…
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    – wilsonw
    Dec 3 '18 at 11:47
















-2












$begingroup$


I am working on a project to determine the Jones polynomial for the torus links and a class of links which I call tst links. Their braid words are respectively given by
$$(sigma_1 sigma_2 cdots sigma_{p-1})^q in B_p$$
and
$$(sigma_1^2 sigma_2 sigma_3 cdots sigma_p)^q in B_{p+1}$$



I would like to know if anyone familiar with the Ocneanu approach, Chern-Simons Theory approach, and the Burau representations approach can evaluate the Jones polynomials and give a clear, detailed explanation.










share|cite|improve this question











$endgroup$












  • $begingroup$
    This is to extend the question math.stackexchange.com/questions/235246/…
    $endgroup$
    – wilsonw
    Dec 3 '18 at 11:47














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$begingroup$


I am working on a project to determine the Jones polynomial for the torus links and a class of links which I call tst links. Their braid words are respectively given by
$$(sigma_1 sigma_2 cdots sigma_{p-1})^q in B_p$$
and
$$(sigma_1^2 sigma_2 sigma_3 cdots sigma_p)^q in B_{p+1}$$



I would like to know if anyone familiar with the Ocneanu approach, Chern-Simons Theory approach, and the Burau representations approach can evaluate the Jones polynomials and give a clear, detailed explanation.










share|cite|improve this question











$endgroup$




I am working on a project to determine the Jones polynomial for the torus links and a class of links which I call tst links. Their braid words are respectively given by
$$(sigma_1 sigma_2 cdots sigma_{p-1})^q in B_p$$
and
$$(sigma_1^2 sigma_2 sigma_3 cdots sigma_p)^q in B_{p+1}$$



I would like to know if anyone familiar with the Ocneanu approach, Chern-Simons Theory approach, and the Burau representations approach can evaluate the Jones polynomials and give a clear, detailed explanation.







knot-theory low-dimensional-topology knot-invariants






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edited Jan 2 at 13:31







wilsonw

















asked Dec 3 '18 at 11:23









wilsonwwilsonw

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467315












  • $begingroup$
    This is to extend the question math.stackexchange.com/questions/235246/…
    $endgroup$
    – wilsonw
    Dec 3 '18 at 11:47


















  • $begingroup$
    This is to extend the question math.stackexchange.com/questions/235246/…
    $endgroup$
    – wilsonw
    Dec 3 '18 at 11:47
















$begingroup$
This is to extend the question math.stackexchange.com/questions/235246/…
$endgroup$
– wilsonw
Dec 3 '18 at 11:47




$begingroup$
This is to extend the question math.stackexchange.com/questions/235246/…
$endgroup$
– wilsonw
Dec 3 '18 at 11:47










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Representation theory approach



First note that the trace of a braid $b$ from $B_p$ is given by the sum over all irreducible representations of the symmetric group $S_p$ given by the Young tableux $Y$ of $p$ nodes:
$$text{tr}(b) = sum_Y tilde W_Y(q,lambda) text{tr}(pi_Y(b))$$
where $tilde W_Y(q,lambda)$ is the weight of the tableau $Y$ and $pi_Y(b)$ is the representation of $b$ corresponding to the tableau $Y$.



Then it is given that $text{tr}(pi_Y(b)) = 0$ unless $Y$ is of the form with $(beta+1)$ boxes in the first row, then $1$ box in each of the $gamma$ rows that follow:



$Y_{beta,gamma}=$Young tableau



The weight of the tableau of this form is given by
$$tilde W_Y(q,lambda) = left(dfrac{1-q}{1-lambda q} right)^p dfrac{R_Y(q,lambda)}{Q_Y(q)}$$
where
$$R_Y(q,lambda) = (1-lambda q)(q-lambda q)(q^2-lambda q) cdots (q^beta -lambda q) times (1-lambda q^2)(1-lambda q^3)cdots (1-lambda q^{gamma+1})$$
$$ = prod_{i=0}^beta (q^i-lambda q)times q^{1+2+cdots+gamma}(q^{-1}-lambda q)(q^{-2}-lambda q)cdots(q^{-gamma}-lambda q)
= q^{1+2+cdots+gamma}prod_{i=-gamma}^beta (q^i-lambda q)
= q^{frac{gamma(gamma+1)}{2}}prod_{i=-gamma}^beta (q^i-lambda q)$$



and $Q_Y$ is given by
$$Q_Y(q) = (1-q)(1-q^2)cdots (1-q^beta) times (1-q^gamma)(1-q^{gamma-1})cdots (1-q) times (1-q^{gamma+beta+1})$$
Using the notation $[n] = 1-q^n$, $[n]! = [n][n-1]!$ and $[0]!=1$, we can write it as
$$Q_Y(q) = [beta]![gamma]!(1-q^p)$$
The remaining piece of information needed to calculate the trace is the trace $text{tr}(pi_Y(b))$ for the Young tableaux $Y = Y_{beta,gamma}$ specified above and $b=(sigma_1sigma_2cdotssigma_{p-1})^m$.



For coprime $m$ and $p$, this is given by
$$text{tr}(pi_Y((sigma_1sigma_2cdotssigma_{p-1})^m) = (-1)^gamma q^{beta m}$$



This is related to the fact that $(sigma_1sigma_2cdotssigma_{p-1})^q$ is in the center of the braid group $B_p$ but I don't know (1) why there is such connection, and (2) why this braid word is in the center.



Putting the pieces of information together, we have
$$text{tr}((sigma_1sigma_2cdotssigma_{p-1})^m)
= sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^gamma q^{beta m}
left(dfrac{1-q}{1-lambda q} right)^p dfrac{q^{frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q) $$



Hence the 2-variable Jones polynomial is given by
$$X_{text{Cl}((sigma_1sigma_2cdotssigma_{p-1})^m)}(q, lambda)
= left(-dfrac{1-lambda q}{sqrt lambda (1-q)}right)^{p-1} sqrt lambda^{(p-1)m} times
sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^gamma q^{beta m}
left(dfrac{1-q}{1-lambda q} right)^p dfrac{q^{frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q)$$

$$= left(dfrac{1-q}{1-lambda q} right) lambda^{frac{(p-1)(m-1)}{2}}
sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^{p-1-gamma} q^{beta m}
dfrac{q^{frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q) $$

$$= left(dfrac{1-q}{1-lambda q} right) lambda^{frac{(p-1)(m-1)}{2}}
sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^beta
dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q) $$





Chern-Simons Theory approach



For a general torus link $K(p,m)$, it can be viewed as a $d$-component link, where $d=text{gcd}(p,m)$, with each component being a $(P,M)$-torus knot, where $dfrac{P}{M}$ is the reduced form of the fraction $dfrac{p}{m}$. We can then split the manifold $S^3$ into pieces with toroidal boundaries such that each piece contains one $K(P,M)$, schematically shown below:
Splitting of the torus link
Then by the spirit of the partition function, we have
$$V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{p-1})^m)}(q, lambda) = left(V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{P-1})^M)}(q, lambda) right)^d
=left( left(dfrac{1-q}{1-lambda q} right) lambda^{frac{(p-d)(m-d)}{2d^2}}
sum_{beta+gamma+1=frac pd; beta,gamma geq 0} (-1)^beta
dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^frac pd)}prod_{i=-gamma}^beta (q^i-lambda q) right)^d
= left(dfrac{1-q}{1-lambda q} right)^d lambda^{frac{(p-d)(m-d)}{2d}}
left(sum_{beta+gamma+1=frac pd; beta,gamma geq 0} (-1)^beta
dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^frac pd)}prod_{i=-gamma}^beta (q^i-lambda q) right)^d$$



Similarly, for the tst links, we have the following splitting of $S^3$ into $(d+1)$ pieces:
Splitting of the tst link
where the rightmost knot is the unknot $O$.



Thus we have
$$V_{text{Cl}((sigma_1^2 sigma_2 cdots sigma_p)^m)}(q, lambda) = left(V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{P-1})^M)}(q, lambda) right)^d V_{O}(q, lambda)
= left(V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{P-1})^M)}(q, lambda) right)^d
= left(dfrac{1-q}{1-lambda q} right)^d lambda^{frac{(p-d)(m-d)}{2d}}
left(sum_{beta+gamma+1=frac pd; beta,gamma geq 0} (-1)^beta
dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^frac pd)}prod_{i=-gamma}^beta (q^i-lambda q) right)^d$$





Ocneanu trace approach



First consider the torus links. I will start with the easiest case $sigma_1 sigma_2 cdots sigma_p$.
The Ocneanu trace is obviously $z^n$.



Then I will try $(sigma_1 sigma_2 cdots sigma_p)^2$. Denote $beta sim_O beta'$ if they have the same Ocneanu traces, i.e. $text{tr}(beta) = text{tr}(beta')$, and $beta sim_M beta'$ if they are related by a sequence of Markov moves. We have
$$(sigma_1 sigma_2 cdots sigma_p)^2 $$
$$= sigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_p $$
$$= sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_p sigma_{p-1} sigma_p $$
$$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1} sigma_p sigma_{p-1} $$
$$sim_O z sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}^2 $$
$$sim_O z(q-1) sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1} + zq sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}$$
$$sim_O z(q-1) (sigma_1 sigma_2 cdots sigma_{p-1})^2 + z^2q sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-2}$$
$$sim_O z(q-1) (sigma_1 sigma_2 cdots sigma_{p-1})^2 + z^2q (sigma_1 sigma_2 cdots sigma_{p-2})^2$$
Writing $tr((sigma_1 sigma_2 cdots sigma_p)^2) = T(2,p)$, we have the recurrence relation
$$T(2,p) = z(q-1) T(2,p-1) + z^2qT(2,p-2)$$
which is second-order, homogeneous and has constant coefficients (no dependance on $p$). The characteristic equation is given by
$$lambda^2 - z(q-1) lambda - z^2q = 0$$
Solving the quadratic equation gives
$$lambda = dfrac{z(q-1) pm sqrt{(z(q-1))^2 - 4(-z^2q)}}{2}
= dfrac{z(q-1) pm sqrt{z^2q^2 - 2z^2q + z^2 + 4z^2q}}{2}
= dfrac{z(q-1) pm sqrt{z^2(q + 1)^2}}{2}
= dfrac{z(q-1) pm z(q+1)}{2}$$

We have $lambda = zq$ or $-z$.
Hence
$$T(2,p) = a(zq)^p + b(-z)^p$$ for some $a$ and $b$.



The initial values are given by
$$T(2,1) = text{tr}(sigma_1^2) = (q-1)text{tr}(sigma_1) + q = (q-1)z + q$$
and
$$T(2,2) = text{tr}(sigma_1sigma_2sigma_1sigma_2) = text{tr}(sigma_1^2sigma_2sigma_1) = ztext{tr}(sigma_1^3) = z(q-1)text{tr}(sigma_1^2) + zq text{tr}(sigma_1) = z(q-1)((q-1)z + q) + z^2q = z^2(q-1)^2 +zq(q-1) + z^2q$$
Substitution gives
$$azq - bz = (q-1)z + q$$
$$a(zq)^2 + bz^2 = z^2(q-1)^2 +zq(q-1) + z^2q$$
Cramer's rule gives
$$a =
dfrac{begin{vmatrix}
(q-1)z + q & -z \
z^2(q-1)^2 +zq(q-1) + z^2q & z^2
end{vmatrix}}
{begin{vmatrix}
zq & -z \
(zq)^2 + zq & z^2
end{vmatrix}}
=dfrac{zbegin{vmatrix}
(q-1)z + q & -1 \
z^2(q-1)^2 +zq(q-1) + z^2q & z
end{vmatrix}}
{z^2qbegin{vmatrix}
1 & -1 \
zq + 1 & z
end{vmatrix}}
=dfrac{(q-1)z^2 + qz + z^2(q-1)^2 +zq(q-1) + z^2q}{zq(z+zq+1)}
=dfrac{zq + z^2q(q-1) +zq(q-1) + z^2q}{zq(z+zq+1)}
=dfrac{q(z + 1)}{z+zq+1}
$$

and
$$b =
dfrac{begin{vmatrix}
zq & (q-1)z + q \
(zq)^2 & z^2(2q-1) + zq
end{vmatrix}}
{begin{vmatrix}
zq & -z \
(zq)^2 + zq & z^2
end{vmatrix}}
=dfrac{zqbegin{vmatrix}
1 & (q-1)z + q \
zq & z^2(q-1)^2 +zq(q-1) + z^2q
end{vmatrix}}{z^2q(z+zq+1)}
=dfrac{zq (z^2(q-1)^2 +zq(q-1) + z^2q - ((q-1)z^2q + zq^2))}{z^2q(z+zq+1)}
=dfrac{z(z-q)}{z(z+zq+1)}
=dfrac{z - q}{z+zq+1}
$$

Hence
$$text{tr}((sigma_1 sigma_2 cdots sigma_p)^2) = dfrac{q(z + 1)(zq)^p}{z+zq+1} + dfrac{(z - q)(-z)^p}{z+zq+1} = dfrac{q(z + 1)(zq)^p + (z - q)(-z)^p}{z+zq+1}$$



Now we try $(sigma_1 sigma_2 cdots sigma_p)^3$. By the same token, we have
$$sigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_p$$
$$=sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_psigma_{p-1}sigma_psigma_1 sigma_2 cdots sigma_p$$
$$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}sigma_psigma_{p-1}sigma_1 sigma_2 cdots sigma_p$$
$$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}^2sigma_1 sigma_2 cdots sigma_{p-1}sigma_psigma_{p-1}$$
$$sim_O zsigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}^2sigma_1 sigma_2 cdots sigma_{p-1}^2$$
$$sim_O z(q-1)(sigma_1 sigma_2 cdots sigma_{p-1})^3 + zqsigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1}^2$$
$$sim_O z(q-1)(sigma_1 sigma_2 cdots sigma_{p-1})^3 + zq(q-1)sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1} + zq^2sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-2}$$
$$sim_O z(q-1)(sigma_1 sigma_2 cdots sigma_{p-1})^3 + zq(q-1)sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1} + z^2q^2(sigma_1 sigma_2 cdots sigma_{p-2})^3$$



But
$$sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1} $$
$$=sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-2}sigma_{p-3}sigma_{p-2} sigma_{p-1}$$
$$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_{p-3} sigma_{p-1}$$
$$= sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-1}sigma_{p-2}sigma_{p-1}sigma_{p-3}$$
$$sim_M sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_{p-1}sigma_{p-2}sigma_{p-3}$$
$$sim_O zsigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}^2sigma_{p-3}$$
$$sim_O z ((q-1) sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_{p-3} + qsigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
$$sim_O z ((q-1) sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-2}sigma_{p-3}sigma_{p-2} + qz(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
$$sim_O z ((q-1) sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2} + qz(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
$$= z ((q-1) (sigma_1 sigma_2 cdots sigma_{p-2})^3 + qz(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
$$sim_O z(q-1) (sigma_1 sigma_2 cdots sigma_{p-2})^3
+ q(q-1)z^2(sigma_1 sigma_2 cdots sigma_{p-3})^3
+q^2z^2(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-4}$$

$$sim_M z(q-1) (sigma_1 sigma_2 cdots sigma_{p-2})^3
+ q(q-1)z^2(sigma_1 sigma_2 cdots sigma_{p-3})^3
+q^2z^2 sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-4} sigma_1 sigma_2 cdots sigma_{p-3}$$

Writing $text{tr}((sigma_1 sigma_2 cdots sigma_p)^3) = T(3, p)$ and
$text{tr}(sigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_p) = U(3, p)$, we have
$$U(3, p-1) = z(q-1)T(3,p-2)+q(q-1)z^2T(3,p-3)+q^2z^2U(3,p-3)$$
and
$$T(3,p) = z(q-1)T(3,p-1) + zq(q-1)U(3,p-1) + z^2q^2T(3,p-2)$$
Rearrange the last relation to give
$$U(3,p-1) = dfrac{T(3,p)-z(q-1)T(3,p-1)-z^2q^2T(3,p-2)}{zq(q-1)}
=dfrac{T(3,p)}{zq(q-1)}-dfrac{T(3,p-1)}{q}-dfrac{zqT(3,p-2)}{q-1}$$

Substitute into the first relation to give
$$dfrac{T(3,p)}{zq(q-1)}-dfrac{T(3,p-1)}{q}-dfrac{zqT(3,p-2)}{q-1}= z(q-1)T(3,p-2)+q(q-1)z^2T(3,p-3)+q^2z^2left(dfrac{T(3,p-2)}{zq(q-1)}-dfrac{T(3,p-3)}{q}-dfrac{zqT(3,p-4)}{q-1}right)$$



This recurrence relation needs to be solved subject to
$$U(3, 2) = text{tr}(sigma_2sigma_1sigma_2) = text{tr}(sigma_1sigma_2sigma_1) = ztext{tr}(sigma_1^2) = z(q-1)text{tr}(sigma_1)+zq = z^2(q-1)+zq$$
$$T(3, 1) = text{tr}(sigma_1^3) = (q-1)text{tr}(sigma_1^2)+qtext{tr}(sigma_1) = (q-1)^2text{tr}(sigma_1)+ (q-1)q + qz
= (q-1)^2z + (q-1)q + qz$$

$$T(3, 2) = text{tr}((sigma_1sigma_2)^3) = text{tr}(sigma_1sigma_2sigma_1sigma_2sigma_1sigma_2) = text{tr}(sigma_2sigma_1sigma_2^2sigma_1sigma_2)
= (q-1)text{tr}(sigma_2sigma_1sigma_2sigma_1sigma_2) + qtext{tr}(sigma_2sigma_1sigma_1^2sigma_2)
= (q-1)text{tr}(sigma_2sigma_1sigma_1sigma_2sigma_1) + q(q-1)text{tr}(sigma_2sigma_1sigma_1sigma_2) + q^2text{tr}(sigma_2sigma_1sigma_2)
= (q-1)^2text{tr}(sigma_2sigma_1sigma_1sigma_2sigma_1) + (q-1)qtext{tr}(sigma_2sigma_2sigma_1)
+ q(q-1)text{tr}(sigma_2^2sigma_1sigma_1) + q^2text{tr}(sigma_1sigma_2sigma_1)
= (q-1)^2text{tr}(sigma_1sigma_1sigma_2sigma_1sigma_2) + (q-1)qtext{tr}(sigma_2sigma_2sigma_1)
+ q(q-1)((q-1)text{tr}(sigma_2sigma_1^2)+qtext{tr}(sigma_1sigma_1))+ q^2ztext{tr}(sigma_1^2)
= (q-1)^2text{tr}(sigma_1sigma_1sigma_1sigma_2sigma_1) + (q-1)qtext{tr}(sigma_1sigma_2sigma_1)
+ q(q-1)((q-1)ztext{tr}(sigma_1^2)+qtr(sigma_1^2))+ q^2ztext{tr}(sigma_1^2)
= (q-1)^2ztext{tr}(sigma_1^4) + (q-1)qztext{tr}(sigma_1^2)
+ q(q-1)((q-1)ztext{tr}(sigma_1^2)+qtr(sigma_1^2))+ q^2ztext{tr}(sigma_1^2)
= (q-1)^2ztext{tr}(sigma_1^4) + ((q-1)qz + q(q-1)((q-1)z+q)+ q^2z)text{tr}(sigma_1^2)$$

But
$$sigma_1^2 sim_O (q-1)sigma_1 + q sim_O (q-1)z + q = qz-z+q$$
and
$$sigma_1^4 sim_O (q-1)sigma_1^3 + qsigma_1^2
sim_O (q-1)((q-1)sigma_1^2 + qsigma_1) + q((q-1)z + q)
sim_O (q-1)((q-1)((q-1)z + q) + qz) + q((q-1)z + q)$$

$$=q^3 z - q^2 z + q z - z + q^3 - q^2 + q$$
Hence
$$T(3, 2) = (q-1)^2z(q-1)((q-1)((q-1)z + q) + qz) + q((q-1)z + q) + ((q-1)qz + q(q-1)((q-1)z+q)+ q^2z)((q-1)z + q)$$
$$=q^5 z^2 + q^5 z - 3 q^4 z^2 - 2 q^4 z + q^4 + 6 q^3 z^2 + 4 q^3 z - q^3 - 7 q^2 z^2 - 2 q^2 z + q^2 + 4 q z^2 - z^2$$






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    $begingroup$

    Representation theory approach



    First note that the trace of a braid $b$ from $B_p$ is given by the sum over all irreducible representations of the symmetric group $S_p$ given by the Young tableux $Y$ of $p$ nodes:
    $$text{tr}(b) = sum_Y tilde W_Y(q,lambda) text{tr}(pi_Y(b))$$
    where $tilde W_Y(q,lambda)$ is the weight of the tableau $Y$ and $pi_Y(b)$ is the representation of $b$ corresponding to the tableau $Y$.



    Then it is given that $text{tr}(pi_Y(b)) = 0$ unless $Y$ is of the form with $(beta+1)$ boxes in the first row, then $1$ box in each of the $gamma$ rows that follow:



    $Y_{beta,gamma}=$Young tableau



    The weight of the tableau of this form is given by
    $$tilde W_Y(q,lambda) = left(dfrac{1-q}{1-lambda q} right)^p dfrac{R_Y(q,lambda)}{Q_Y(q)}$$
    where
    $$R_Y(q,lambda) = (1-lambda q)(q-lambda q)(q^2-lambda q) cdots (q^beta -lambda q) times (1-lambda q^2)(1-lambda q^3)cdots (1-lambda q^{gamma+1})$$
    $$ = prod_{i=0}^beta (q^i-lambda q)times q^{1+2+cdots+gamma}(q^{-1}-lambda q)(q^{-2}-lambda q)cdots(q^{-gamma}-lambda q)
    = q^{1+2+cdots+gamma}prod_{i=-gamma}^beta (q^i-lambda q)
    = q^{frac{gamma(gamma+1)}{2}}prod_{i=-gamma}^beta (q^i-lambda q)$$



    and $Q_Y$ is given by
    $$Q_Y(q) = (1-q)(1-q^2)cdots (1-q^beta) times (1-q^gamma)(1-q^{gamma-1})cdots (1-q) times (1-q^{gamma+beta+1})$$
    Using the notation $[n] = 1-q^n$, $[n]! = [n][n-1]!$ and $[0]!=1$, we can write it as
    $$Q_Y(q) = [beta]![gamma]!(1-q^p)$$
    The remaining piece of information needed to calculate the trace is the trace $text{tr}(pi_Y(b))$ for the Young tableaux $Y = Y_{beta,gamma}$ specified above and $b=(sigma_1sigma_2cdotssigma_{p-1})^m$.



    For coprime $m$ and $p$, this is given by
    $$text{tr}(pi_Y((sigma_1sigma_2cdotssigma_{p-1})^m) = (-1)^gamma q^{beta m}$$



    This is related to the fact that $(sigma_1sigma_2cdotssigma_{p-1})^q$ is in the center of the braid group $B_p$ but I don't know (1) why there is such connection, and (2) why this braid word is in the center.



    Putting the pieces of information together, we have
    $$text{tr}((sigma_1sigma_2cdotssigma_{p-1})^m)
    = sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^gamma q^{beta m}
    left(dfrac{1-q}{1-lambda q} right)^p dfrac{q^{frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q) $$



    Hence the 2-variable Jones polynomial is given by
    $$X_{text{Cl}((sigma_1sigma_2cdotssigma_{p-1})^m)}(q, lambda)
    = left(-dfrac{1-lambda q}{sqrt lambda (1-q)}right)^{p-1} sqrt lambda^{(p-1)m} times
    sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^gamma q^{beta m}
    left(dfrac{1-q}{1-lambda q} right)^p dfrac{q^{frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q)$$

    $$= left(dfrac{1-q}{1-lambda q} right) lambda^{frac{(p-1)(m-1)}{2}}
    sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^{p-1-gamma} q^{beta m}
    dfrac{q^{frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q) $$

    $$= left(dfrac{1-q}{1-lambda q} right) lambda^{frac{(p-1)(m-1)}{2}}
    sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^beta
    dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q) $$





    Chern-Simons Theory approach



    For a general torus link $K(p,m)$, it can be viewed as a $d$-component link, where $d=text{gcd}(p,m)$, with each component being a $(P,M)$-torus knot, where $dfrac{P}{M}$ is the reduced form of the fraction $dfrac{p}{m}$. We can then split the manifold $S^3$ into pieces with toroidal boundaries such that each piece contains one $K(P,M)$, schematically shown below:
    Splitting of the torus link
    Then by the spirit of the partition function, we have
    $$V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{p-1})^m)}(q, lambda) = left(V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{P-1})^M)}(q, lambda) right)^d
    =left( left(dfrac{1-q}{1-lambda q} right) lambda^{frac{(p-d)(m-d)}{2d^2}}
    sum_{beta+gamma+1=frac pd; beta,gamma geq 0} (-1)^beta
    dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^frac pd)}prod_{i=-gamma}^beta (q^i-lambda q) right)^d
    = left(dfrac{1-q}{1-lambda q} right)^d lambda^{frac{(p-d)(m-d)}{2d}}
    left(sum_{beta+gamma+1=frac pd; beta,gamma geq 0} (-1)^beta
    dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^frac pd)}prod_{i=-gamma}^beta (q^i-lambda q) right)^d$$



    Similarly, for the tst links, we have the following splitting of $S^3$ into $(d+1)$ pieces:
    Splitting of the tst link
    where the rightmost knot is the unknot $O$.



    Thus we have
    $$V_{text{Cl}((sigma_1^2 sigma_2 cdots sigma_p)^m)}(q, lambda) = left(V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{P-1})^M)}(q, lambda) right)^d V_{O}(q, lambda)
    = left(V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{P-1})^M)}(q, lambda) right)^d
    = left(dfrac{1-q}{1-lambda q} right)^d lambda^{frac{(p-d)(m-d)}{2d}}
    left(sum_{beta+gamma+1=frac pd; beta,gamma geq 0} (-1)^beta
    dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^frac pd)}prod_{i=-gamma}^beta (q^i-lambda q) right)^d$$





    Ocneanu trace approach



    First consider the torus links. I will start with the easiest case $sigma_1 sigma_2 cdots sigma_p$.
    The Ocneanu trace is obviously $z^n$.



    Then I will try $(sigma_1 sigma_2 cdots sigma_p)^2$. Denote $beta sim_O beta'$ if they have the same Ocneanu traces, i.e. $text{tr}(beta) = text{tr}(beta')$, and $beta sim_M beta'$ if they are related by a sequence of Markov moves. We have
    $$(sigma_1 sigma_2 cdots sigma_p)^2 $$
    $$= sigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_p $$
    $$= sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_p sigma_{p-1} sigma_p $$
    $$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1} sigma_p sigma_{p-1} $$
    $$sim_O z sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}^2 $$
    $$sim_O z(q-1) sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1} + zq sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}$$
    $$sim_O z(q-1) (sigma_1 sigma_2 cdots sigma_{p-1})^2 + z^2q sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-2}$$
    $$sim_O z(q-1) (sigma_1 sigma_2 cdots sigma_{p-1})^2 + z^2q (sigma_1 sigma_2 cdots sigma_{p-2})^2$$
    Writing $tr((sigma_1 sigma_2 cdots sigma_p)^2) = T(2,p)$, we have the recurrence relation
    $$T(2,p) = z(q-1) T(2,p-1) + z^2qT(2,p-2)$$
    which is second-order, homogeneous and has constant coefficients (no dependance on $p$). The characteristic equation is given by
    $$lambda^2 - z(q-1) lambda - z^2q = 0$$
    Solving the quadratic equation gives
    $$lambda = dfrac{z(q-1) pm sqrt{(z(q-1))^2 - 4(-z^2q)}}{2}
    = dfrac{z(q-1) pm sqrt{z^2q^2 - 2z^2q + z^2 + 4z^2q}}{2}
    = dfrac{z(q-1) pm sqrt{z^2(q + 1)^2}}{2}
    = dfrac{z(q-1) pm z(q+1)}{2}$$

    We have $lambda = zq$ or $-z$.
    Hence
    $$T(2,p) = a(zq)^p + b(-z)^p$$ for some $a$ and $b$.



    The initial values are given by
    $$T(2,1) = text{tr}(sigma_1^2) = (q-1)text{tr}(sigma_1) + q = (q-1)z + q$$
    and
    $$T(2,2) = text{tr}(sigma_1sigma_2sigma_1sigma_2) = text{tr}(sigma_1^2sigma_2sigma_1) = ztext{tr}(sigma_1^3) = z(q-1)text{tr}(sigma_1^2) + zq text{tr}(sigma_1) = z(q-1)((q-1)z + q) + z^2q = z^2(q-1)^2 +zq(q-1) + z^2q$$
    Substitution gives
    $$azq - bz = (q-1)z + q$$
    $$a(zq)^2 + bz^2 = z^2(q-1)^2 +zq(q-1) + z^2q$$
    Cramer's rule gives
    $$a =
    dfrac{begin{vmatrix}
    (q-1)z + q & -z \
    z^2(q-1)^2 +zq(q-1) + z^2q & z^2
    end{vmatrix}}
    {begin{vmatrix}
    zq & -z \
    (zq)^2 + zq & z^2
    end{vmatrix}}
    =dfrac{zbegin{vmatrix}
    (q-1)z + q & -1 \
    z^2(q-1)^2 +zq(q-1) + z^2q & z
    end{vmatrix}}
    {z^2qbegin{vmatrix}
    1 & -1 \
    zq + 1 & z
    end{vmatrix}}
    =dfrac{(q-1)z^2 + qz + z^2(q-1)^2 +zq(q-1) + z^2q}{zq(z+zq+1)}
    =dfrac{zq + z^2q(q-1) +zq(q-1) + z^2q}{zq(z+zq+1)}
    =dfrac{q(z + 1)}{z+zq+1}
    $$

    and
    $$b =
    dfrac{begin{vmatrix}
    zq & (q-1)z + q \
    (zq)^2 & z^2(2q-1) + zq
    end{vmatrix}}
    {begin{vmatrix}
    zq & -z \
    (zq)^2 + zq & z^2
    end{vmatrix}}
    =dfrac{zqbegin{vmatrix}
    1 & (q-1)z + q \
    zq & z^2(q-1)^2 +zq(q-1) + z^2q
    end{vmatrix}}{z^2q(z+zq+1)}
    =dfrac{zq (z^2(q-1)^2 +zq(q-1) + z^2q - ((q-1)z^2q + zq^2))}{z^2q(z+zq+1)}
    =dfrac{z(z-q)}{z(z+zq+1)}
    =dfrac{z - q}{z+zq+1}
    $$

    Hence
    $$text{tr}((sigma_1 sigma_2 cdots sigma_p)^2) = dfrac{q(z + 1)(zq)^p}{z+zq+1} + dfrac{(z - q)(-z)^p}{z+zq+1} = dfrac{q(z + 1)(zq)^p + (z - q)(-z)^p}{z+zq+1}$$



    Now we try $(sigma_1 sigma_2 cdots sigma_p)^3$. By the same token, we have
    $$sigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_p$$
    $$=sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_psigma_{p-1}sigma_psigma_1 sigma_2 cdots sigma_p$$
    $$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}sigma_psigma_{p-1}sigma_1 sigma_2 cdots sigma_p$$
    $$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}^2sigma_1 sigma_2 cdots sigma_{p-1}sigma_psigma_{p-1}$$
    $$sim_O zsigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}^2sigma_1 sigma_2 cdots sigma_{p-1}^2$$
    $$sim_O z(q-1)(sigma_1 sigma_2 cdots sigma_{p-1})^3 + zqsigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1}^2$$
    $$sim_O z(q-1)(sigma_1 sigma_2 cdots sigma_{p-1})^3 + zq(q-1)sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1} + zq^2sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-2}$$
    $$sim_O z(q-1)(sigma_1 sigma_2 cdots sigma_{p-1})^3 + zq(q-1)sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1} + z^2q^2(sigma_1 sigma_2 cdots sigma_{p-2})^3$$



    But
    $$sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1} $$
    $$=sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-2}sigma_{p-3}sigma_{p-2} sigma_{p-1}$$
    $$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_{p-3} sigma_{p-1}$$
    $$= sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-1}sigma_{p-2}sigma_{p-1}sigma_{p-3}$$
    $$sim_M sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_{p-1}sigma_{p-2}sigma_{p-3}$$
    $$sim_O zsigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}^2sigma_{p-3}$$
    $$sim_O z ((q-1) sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_{p-3} + qsigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
    $$sim_O z ((q-1) sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-2}sigma_{p-3}sigma_{p-2} + qz(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
    $$sim_O z ((q-1) sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2} + qz(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
    $$= z ((q-1) (sigma_1 sigma_2 cdots sigma_{p-2})^3 + qz(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
    $$sim_O z(q-1) (sigma_1 sigma_2 cdots sigma_{p-2})^3
    + q(q-1)z^2(sigma_1 sigma_2 cdots sigma_{p-3})^3
    +q^2z^2(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-4}$$

    $$sim_M z(q-1) (sigma_1 sigma_2 cdots sigma_{p-2})^3
    + q(q-1)z^2(sigma_1 sigma_2 cdots sigma_{p-3})^3
    +q^2z^2 sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-4} sigma_1 sigma_2 cdots sigma_{p-3}$$

    Writing $text{tr}((sigma_1 sigma_2 cdots sigma_p)^3) = T(3, p)$ and
    $text{tr}(sigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_p) = U(3, p)$, we have
    $$U(3, p-1) = z(q-1)T(3,p-2)+q(q-1)z^2T(3,p-3)+q^2z^2U(3,p-3)$$
    and
    $$T(3,p) = z(q-1)T(3,p-1) + zq(q-1)U(3,p-1) + z^2q^2T(3,p-2)$$
    Rearrange the last relation to give
    $$U(3,p-1) = dfrac{T(3,p)-z(q-1)T(3,p-1)-z^2q^2T(3,p-2)}{zq(q-1)}
    =dfrac{T(3,p)}{zq(q-1)}-dfrac{T(3,p-1)}{q}-dfrac{zqT(3,p-2)}{q-1}$$

    Substitute into the first relation to give
    $$dfrac{T(3,p)}{zq(q-1)}-dfrac{T(3,p-1)}{q}-dfrac{zqT(3,p-2)}{q-1}= z(q-1)T(3,p-2)+q(q-1)z^2T(3,p-3)+q^2z^2left(dfrac{T(3,p-2)}{zq(q-1)}-dfrac{T(3,p-3)}{q}-dfrac{zqT(3,p-4)}{q-1}right)$$



    This recurrence relation needs to be solved subject to
    $$U(3, 2) = text{tr}(sigma_2sigma_1sigma_2) = text{tr}(sigma_1sigma_2sigma_1) = ztext{tr}(sigma_1^2) = z(q-1)text{tr}(sigma_1)+zq = z^2(q-1)+zq$$
    $$T(3, 1) = text{tr}(sigma_1^3) = (q-1)text{tr}(sigma_1^2)+qtext{tr}(sigma_1) = (q-1)^2text{tr}(sigma_1)+ (q-1)q + qz
    = (q-1)^2z + (q-1)q + qz$$

    $$T(3, 2) = text{tr}((sigma_1sigma_2)^3) = text{tr}(sigma_1sigma_2sigma_1sigma_2sigma_1sigma_2) = text{tr}(sigma_2sigma_1sigma_2^2sigma_1sigma_2)
    = (q-1)text{tr}(sigma_2sigma_1sigma_2sigma_1sigma_2) + qtext{tr}(sigma_2sigma_1sigma_1^2sigma_2)
    = (q-1)text{tr}(sigma_2sigma_1sigma_1sigma_2sigma_1) + q(q-1)text{tr}(sigma_2sigma_1sigma_1sigma_2) + q^2text{tr}(sigma_2sigma_1sigma_2)
    = (q-1)^2text{tr}(sigma_2sigma_1sigma_1sigma_2sigma_1) + (q-1)qtext{tr}(sigma_2sigma_2sigma_1)
    + q(q-1)text{tr}(sigma_2^2sigma_1sigma_1) + q^2text{tr}(sigma_1sigma_2sigma_1)
    = (q-1)^2text{tr}(sigma_1sigma_1sigma_2sigma_1sigma_2) + (q-1)qtext{tr}(sigma_2sigma_2sigma_1)
    + q(q-1)((q-1)text{tr}(sigma_2sigma_1^2)+qtext{tr}(sigma_1sigma_1))+ q^2ztext{tr}(sigma_1^2)
    = (q-1)^2text{tr}(sigma_1sigma_1sigma_1sigma_2sigma_1) + (q-1)qtext{tr}(sigma_1sigma_2sigma_1)
    + q(q-1)((q-1)ztext{tr}(sigma_1^2)+qtr(sigma_1^2))+ q^2ztext{tr}(sigma_1^2)
    = (q-1)^2ztext{tr}(sigma_1^4) + (q-1)qztext{tr}(sigma_1^2)
    + q(q-1)((q-1)ztext{tr}(sigma_1^2)+qtr(sigma_1^2))+ q^2ztext{tr}(sigma_1^2)
    = (q-1)^2ztext{tr}(sigma_1^4) + ((q-1)qz + q(q-1)((q-1)z+q)+ q^2z)text{tr}(sigma_1^2)$$

    But
    $$sigma_1^2 sim_O (q-1)sigma_1 + q sim_O (q-1)z + q = qz-z+q$$
    and
    $$sigma_1^4 sim_O (q-1)sigma_1^3 + qsigma_1^2
    sim_O (q-1)((q-1)sigma_1^2 + qsigma_1) + q((q-1)z + q)
    sim_O (q-1)((q-1)((q-1)z + q) + qz) + q((q-1)z + q)$$

    $$=q^3 z - q^2 z + q z - z + q^3 - q^2 + q$$
    Hence
    $$T(3, 2) = (q-1)^2z(q-1)((q-1)((q-1)z + q) + qz) + q((q-1)z + q) + ((q-1)qz + q(q-1)((q-1)z+q)+ q^2z)((q-1)z + q)$$
    $$=q^5 z^2 + q^5 z - 3 q^4 z^2 - 2 q^4 z + q^4 + 6 q^3 z^2 + 4 q^3 z - q^3 - 7 q^2 z^2 - 2 q^2 z + q^2 + 4 q z^2 - z^2$$






    share|cite|improve this answer











    $endgroup$


















      4












      $begingroup$

      Representation theory approach



      First note that the trace of a braid $b$ from $B_p$ is given by the sum over all irreducible representations of the symmetric group $S_p$ given by the Young tableux $Y$ of $p$ nodes:
      $$text{tr}(b) = sum_Y tilde W_Y(q,lambda) text{tr}(pi_Y(b))$$
      where $tilde W_Y(q,lambda)$ is the weight of the tableau $Y$ and $pi_Y(b)$ is the representation of $b$ corresponding to the tableau $Y$.



      Then it is given that $text{tr}(pi_Y(b)) = 0$ unless $Y$ is of the form with $(beta+1)$ boxes in the first row, then $1$ box in each of the $gamma$ rows that follow:



      $Y_{beta,gamma}=$Young tableau



      The weight of the tableau of this form is given by
      $$tilde W_Y(q,lambda) = left(dfrac{1-q}{1-lambda q} right)^p dfrac{R_Y(q,lambda)}{Q_Y(q)}$$
      where
      $$R_Y(q,lambda) = (1-lambda q)(q-lambda q)(q^2-lambda q) cdots (q^beta -lambda q) times (1-lambda q^2)(1-lambda q^3)cdots (1-lambda q^{gamma+1})$$
      $$ = prod_{i=0}^beta (q^i-lambda q)times q^{1+2+cdots+gamma}(q^{-1}-lambda q)(q^{-2}-lambda q)cdots(q^{-gamma}-lambda q)
      = q^{1+2+cdots+gamma}prod_{i=-gamma}^beta (q^i-lambda q)
      = q^{frac{gamma(gamma+1)}{2}}prod_{i=-gamma}^beta (q^i-lambda q)$$



      and $Q_Y$ is given by
      $$Q_Y(q) = (1-q)(1-q^2)cdots (1-q^beta) times (1-q^gamma)(1-q^{gamma-1})cdots (1-q) times (1-q^{gamma+beta+1})$$
      Using the notation $[n] = 1-q^n$, $[n]! = [n][n-1]!$ and $[0]!=1$, we can write it as
      $$Q_Y(q) = [beta]![gamma]!(1-q^p)$$
      The remaining piece of information needed to calculate the trace is the trace $text{tr}(pi_Y(b))$ for the Young tableaux $Y = Y_{beta,gamma}$ specified above and $b=(sigma_1sigma_2cdotssigma_{p-1})^m$.



      For coprime $m$ and $p$, this is given by
      $$text{tr}(pi_Y((sigma_1sigma_2cdotssigma_{p-1})^m) = (-1)^gamma q^{beta m}$$



      This is related to the fact that $(sigma_1sigma_2cdotssigma_{p-1})^q$ is in the center of the braid group $B_p$ but I don't know (1) why there is such connection, and (2) why this braid word is in the center.



      Putting the pieces of information together, we have
      $$text{tr}((sigma_1sigma_2cdotssigma_{p-1})^m)
      = sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^gamma q^{beta m}
      left(dfrac{1-q}{1-lambda q} right)^p dfrac{q^{frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q) $$



      Hence the 2-variable Jones polynomial is given by
      $$X_{text{Cl}((sigma_1sigma_2cdotssigma_{p-1})^m)}(q, lambda)
      = left(-dfrac{1-lambda q}{sqrt lambda (1-q)}right)^{p-1} sqrt lambda^{(p-1)m} times
      sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^gamma q^{beta m}
      left(dfrac{1-q}{1-lambda q} right)^p dfrac{q^{frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q)$$

      $$= left(dfrac{1-q}{1-lambda q} right) lambda^{frac{(p-1)(m-1)}{2}}
      sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^{p-1-gamma} q^{beta m}
      dfrac{q^{frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q) $$

      $$= left(dfrac{1-q}{1-lambda q} right) lambda^{frac{(p-1)(m-1)}{2}}
      sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^beta
      dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q) $$





      Chern-Simons Theory approach



      For a general torus link $K(p,m)$, it can be viewed as a $d$-component link, where $d=text{gcd}(p,m)$, with each component being a $(P,M)$-torus knot, where $dfrac{P}{M}$ is the reduced form of the fraction $dfrac{p}{m}$. We can then split the manifold $S^3$ into pieces with toroidal boundaries such that each piece contains one $K(P,M)$, schematically shown below:
      Splitting of the torus link
      Then by the spirit of the partition function, we have
      $$V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{p-1})^m)}(q, lambda) = left(V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{P-1})^M)}(q, lambda) right)^d
      =left( left(dfrac{1-q}{1-lambda q} right) lambda^{frac{(p-d)(m-d)}{2d^2}}
      sum_{beta+gamma+1=frac pd; beta,gamma geq 0} (-1)^beta
      dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^frac pd)}prod_{i=-gamma}^beta (q^i-lambda q) right)^d
      = left(dfrac{1-q}{1-lambda q} right)^d lambda^{frac{(p-d)(m-d)}{2d}}
      left(sum_{beta+gamma+1=frac pd; beta,gamma geq 0} (-1)^beta
      dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^frac pd)}prod_{i=-gamma}^beta (q^i-lambda q) right)^d$$



      Similarly, for the tst links, we have the following splitting of $S^3$ into $(d+1)$ pieces:
      Splitting of the tst link
      where the rightmost knot is the unknot $O$.



      Thus we have
      $$V_{text{Cl}((sigma_1^2 sigma_2 cdots sigma_p)^m)}(q, lambda) = left(V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{P-1})^M)}(q, lambda) right)^d V_{O}(q, lambda)
      = left(V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{P-1})^M)}(q, lambda) right)^d
      = left(dfrac{1-q}{1-lambda q} right)^d lambda^{frac{(p-d)(m-d)}{2d}}
      left(sum_{beta+gamma+1=frac pd; beta,gamma geq 0} (-1)^beta
      dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^frac pd)}prod_{i=-gamma}^beta (q^i-lambda q) right)^d$$





      Ocneanu trace approach



      First consider the torus links. I will start with the easiest case $sigma_1 sigma_2 cdots sigma_p$.
      The Ocneanu trace is obviously $z^n$.



      Then I will try $(sigma_1 sigma_2 cdots sigma_p)^2$. Denote $beta sim_O beta'$ if they have the same Ocneanu traces, i.e. $text{tr}(beta) = text{tr}(beta')$, and $beta sim_M beta'$ if they are related by a sequence of Markov moves. We have
      $$(sigma_1 sigma_2 cdots sigma_p)^2 $$
      $$= sigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_p $$
      $$= sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_p sigma_{p-1} sigma_p $$
      $$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1} sigma_p sigma_{p-1} $$
      $$sim_O z sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}^2 $$
      $$sim_O z(q-1) sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1} + zq sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}$$
      $$sim_O z(q-1) (sigma_1 sigma_2 cdots sigma_{p-1})^2 + z^2q sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-2}$$
      $$sim_O z(q-1) (sigma_1 sigma_2 cdots sigma_{p-1})^2 + z^2q (sigma_1 sigma_2 cdots sigma_{p-2})^2$$
      Writing $tr((sigma_1 sigma_2 cdots sigma_p)^2) = T(2,p)$, we have the recurrence relation
      $$T(2,p) = z(q-1) T(2,p-1) + z^2qT(2,p-2)$$
      which is second-order, homogeneous and has constant coefficients (no dependance on $p$). The characteristic equation is given by
      $$lambda^2 - z(q-1) lambda - z^2q = 0$$
      Solving the quadratic equation gives
      $$lambda = dfrac{z(q-1) pm sqrt{(z(q-1))^2 - 4(-z^2q)}}{2}
      = dfrac{z(q-1) pm sqrt{z^2q^2 - 2z^2q + z^2 + 4z^2q}}{2}
      = dfrac{z(q-1) pm sqrt{z^2(q + 1)^2}}{2}
      = dfrac{z(q-1) pm z(q+1)}{2}$$

      We have $lambda = zq$ or $-z$.
      Hence
      $$T(2,p) = a(zq)^p + b(-z)^p$$ for some $a$ and $b$.



      The initial values are given by
      $$T(2,1) = text{tr}(sigma_1^2) = (q-1)text{tr}(sigma_1) + q = (q-1)z + q$$
      and
      $$T(2,2) = text{tr}(sigma_1sigma_2sigma_1sigma_2) = text{tr}(sigma_1^2sigma_2sigma_1) = ztext{tr}(sigma_1^3) = z(q-1)text{tr}(sigma_1^2) + zq text{tr}(sigma_1) = z(q-1)((q-1)z + q) + z^2q = z^2(q-1)^2 +zq(q-1) + z^2q$$
      Substitution gives
      $$azq - bz = (q-1)z + q$$
      $$a(zq)^2 + bz^2 = z^2(q-1)^2 +zq(q-1) + z^2q$$
      Cramer's rule gives
      $$a =
      dfrac{begin{vmatrix}
      (q-1)z + q & -z \
      z^2(q-1)^2 +zq(q-1) + z^2q & z^2
      end{vmatrix}}
      {begin{vmatrix}
      zq & -z \
      (zq)^2 + zq & z^2
      end{vmatrix}}
      =dfrac{zbegin{vmatrix}
      (q-1)z + q & -1 \
      z^2(q-1)^2 +zq(q-1) + z^2q & z
      end{vmatrix}}
      {z^2qbegin{vmatrix}
      1 & -1 \
      zq + 1 & z
      end{vmatrix}}
      =dfrac{(q-1)z^2 + qz + z^2(q-1)^2 +zq(q-1) + z^2q}{zq(z+zq+1)}
      =dfrac{zq + z^2q(q-1) +zq(q-1) + z^2q}{zq(z+zq+1)}
      =dfrac{q(z + 1)}{z+zq+1}
      $$

      and
      $$b =
      dfrac{begin{vmatrix}
      zq & (q-1)z + q \
      (zq)^2 & z^2(2q-1) + zq
      end{vmatrix}}
      {begin{vmatrix}
      zq & -z \
      (zq)^2 + zq & z^2
      end{vmatrix}}
      =dfrac{zqbegin{vmatrix}
      1 & (q-1)z + q \
      zq & z^2(q-1)^2 +zq(q-1) + z^2q
      end{vmatrix}}{z^2q(z+zq+1)}
      =dfrac{zq (z^2(q-1)^2 +zq(q-1) + z^2q - ((q-1)z^2q + zq^2))}{z^2q(z+zq+1)}
      =dfrac{z(z-q)}{z(z+zq+1)}
      =dfrac{z - q}{z+zq+1}
      $$

      Hence
      $$text{tr}((sigma_1 sigma_2 cdots sigma_p)^2) = dfrac{q(z + 1)(zq)^p}{z+zq+1} + dfrac{(z - q)(-z)^p}{z+zq+1} = dfrac{q(z + 1)(zq)^p + (z - q)(-z)^p}{z+zq+1}$$



      Now we try $(sigma_1 sigma_2 cdots sigma_p)^3$. By the same token, we have
      $$sigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_p$$
      $$=sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_psigma_{p-1}sigma_psigma_1 sigma_2 cdots sigma_p$$
      $$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}sigma_psigma_{p-1}sigma_1 sigma_2 cdots sigma_p$$
      $$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}^2sigma_1 sigma_2 cdots sigma_{p-1}sigma_psigma_{p-1}$$
      $$sim_O zsigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}^2sigma_1 sigma_2 cdots sigma_{p-1}^2$$
      $$sim_O z(q-1)(sigma_1 sigma_2 cdots sigma_{p-1})^3 + zqsigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1}^2$$
      $$sim_O z(q-1)(sigma_1 sigma_2 cdots sigma_{p-1})^3 + zq(q-1)sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1} + zq^2sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-2}$$
      $$sim_O z(q-1)(sigma_1 sigma_2 cdots sigma_{p-1})^3 + zq(q-1)sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1} + z^2q^2(sigma_1 sigma_2 cdots sigma_{p-2})^3$$



      But
      $$sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1} $$
      $$=sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-2}sigma_{p-3}sigma_{p-2} sigma_{p-1}$$
      $$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_{p-3} sigma_{p-1}$$
      $$= sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-1}sigma_{p-2}sigma_{p-1}sigma_{p-3}$$
      $$sim_M sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_{p-1}sigma_{p-2}sigma_{p-3}$$
      $$sim_O zsigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}^2sigma_{p-3}$$
      $$sim_O z ((q-1) sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_{p-3} + qsigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
      $$sim_O z ((q-1) sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-2}sigma_{p-3}sigma_{p-2} + qz(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
      $$sim_O z ((q-1) sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2} + qz(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
      $$= z ((q-1) (sigma_1 sigma_2 cdots sigma_{p-2})^3 + qz(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
      $$sim_O z(q-1) (sigma_1 sigma_2 cdots sigma_{p-2})^3
      + q(q-1)z^2(sigma_1 sigma_2 cdots sigma_{p-3})^3
      +q^2z^2(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-4}$$

      $$sim_M z(q-1) (sigma_1 sigma_2 cdots sigma_{p-2})^3
      + q(q-1)z^2(sigma_1 sigma_2 cdots sigma_{p-3})^3
      +q^2z^2 sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-4} sigma_1 sigma_2 cdots sigma_{p-3}$$

      Writing $text{tr}((sigma_1 sigma_2 cdots sigma_p)^3) = T(3, p)$ and
      $text{tr}(sigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_p) = U(3, p)$, we have
      $$U(3, p-1) = z(q-1)T(3,p-2)+q(q-1)z^2T(3,p-3)+q^2z^2U(3,p-3)$$
      and
      $$T(3,p) = z(q-1)T(3,p-1) + zq(q-1)U(3,p-1) + z^2q^2T(3,p-2)$$
      Rearrange the last relation to give
      $$U(3,p-1) = dfrac{T(3,p)-z(q-1)T(3,p-1)-z^2q^2T(3,p-2)}{zq(q-1)}
      =dfrac{T(3,p)}{zq(q-1)}-dfrac{T(3,p-1)}{q}-dfrac{zqT(3,p-2)}{q-1}$$

      Substitute into the first relation to give
      $$dfrac{T(3,p)}{zq(q-1)}-dfrac{T(3,p-1)}{q}-dfrac{zqT(3,p-2)}{q-1}= z(q-1)T(3,p-2)+q(q-1)z^2T(3,p-3)+q^2z^2left(dfrac{T(3,p-2)}{zq(q-1)}-dfrac{T(3,p-3)}{q}-dfrac{zqT(3,p-4)}{q-1}right)$$



      This recurrence relation needs to be solved subject to
      $$U(3, 2) = text{tr}(sigma_2sigma_1sigma_2) = text{tr}(sigma_1sigma_2sigma_1) = ztext{tr}(sigma_1^2) = z(q-1)text{tr}(sigma_1)+zq = z^2(q-1)+zq$$
      $$T(3, 1) = text{tr}(sigma_1^3) = (q-1)text{tr}(sigma_1^2)+qtext{tr}(sigma_1) = (q-1)^2text{tr}(sigma_1)+ (q-1)q + qz
      = (q-1)^2z + (q-1)q + qz$$

      $$T(3, 2) = text{tr}((sigma_1sigma_2)^3) = text{tr}(sigma_1sigma_2sigma_1sigma_2sigma_1sigma_2) = text{tr}(sigma_2sigma_1sigma_2^2sigma_1sigma_2)
      = (q-1)text{tr}(sigma_2sigma_1sigma_2sigma_1sigma_2) + qtext{tr}(sigma_2sigma_1sigma_1^2sigma_2)
      = (q-1)text{tr}(sigma_2sigma_1sigma_1sigma_2sigma_1) + q(q-1)text{tr}(sigma_2sigma_1sigma_1sigma_2) + q^2text{tr}(sigma_2sigma_1sigma_2)
      = (q-1)^2text{tr}(sigma_2sigma_1sigma_1sigma_2sigma_1) + (q-1)qtext{tr}(sigma_2sigma_2sigma_1)
      + q(q-1)text{tr}(sigma_2^2sigma_1sigma_1) + q^2text{tr}(sigma_1sigma_2sigma_1)
      = (q-1)^2text{tr}(sigma_1sigma_1sigma_2sigma_1sigma_2) + (q-1)qtext{tr}(sigma_2sigma_2sigma_1)
      + q(q-1)((q-1)text{tr}(sigma_2sigma_1^2)+qtext{tr}(sigma_1sigma_1))+ q^2ztext{tr}(sigma_1^2)
      = (q-1)^2text{tr}(sigma_1sigma_1sigma_1sigma_2sigma_1) + (q-1)qtext{tr}(sigma_1sigma_2sigma_1)
      + q(q-1)((q-1)ztext{tr}(sigma_1^2)+qtr(sigma_1^2))+ q^2ztext{tr}(sigma_1^2)
      = (q-1)^2ztext{tr}(sigma_1^4) + (q-1)qztext{tr}(sigma_1^2)
      + q(q-1)((q-1)ztext{tr}(sigma_1^2)+qtr(sigma_1^2))+ q^2ztext{tr}(sigma_1^2)
      = (q-1)^2ztext{tr}(sigma_1^4) + ((q-1)qz + q(q-1)((q-1)z+q)+ q^2z)text{tr}(sigma_1^2)$$

      But
      $$sigma_1^2 sim_O (q-1)sigma_1 + q sim_O (q-1)z + q = qz-z+q$$
      and
      $$sigma_1^4 sim_O (q-1)sigma_1^3 + qsigma_1^2
      sim_O (q-1)((q-1)sigma_1^2 + qsigma_1) + q((q-1)z + q)
      sim_O (q-1)((q-1)((q-1)z + q) + qz) + q((q-1)z + q)$$

      $$=q^3 z - q^2 z + q z - z + q^3 - q^2 + q$$
      Hence
      $$T(3, 2) = (q-1)^2z(q-1)((q-1)((q-1)z + q) + qz) + q((q-1)z + q) + ((q-1)qz + q(q-1)((q-1)z+q)+ q^2z)((q-1)z + q)$$
      $$=q^5 z^2 + q^5 z - 3 q^4 z^2 - 2 q^4 z + q^4 + 6 q^3 z^2 + 4 q^3 z - q^3 - 7 q^2 z^2 - 2 q^2 z + q^2 + 4 q z^2 - z^2$$






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      $endgroup$
















        4












        4








        4





        $begingroup$

        Representation theory approach



        First note that the trace of a braid $b$ from $B_p$ is given by the sum over all irreducible representations of the symmetric group $S_p$ given by the Young tableux $Y$ of $p$ nodes:
        $$text{tr}(b) = sum_Y tilde W_Y(q,lambda) text{tr}(pi_Y(b))$$
        where $tilde W_Y(q,lambda)$ is the weight of the tableau $Y$ and $pi_Y(b)$ is the representation of $b$ corresponding to the tableau $Y$.



        Then it is given that $text{tr}(pi_Y(b)) = 0$ unless $Y$ is of the form with $(beta+1)$ boxes in the first row, then $1$ box in each of the $gamma$ rows that follow:



        $Y_{beta,gamma}=$Young tableau



        The weight of the tableau of this form is given by
        $$tilde W_Y(q,lambda) = left(dfrac{1-q}{1-lambda q} right)^p dfrac{R_Y(q,lambda)}{Q_Y(q)}$$
        where
        $$R_Y(q,lambda) = (1-lambda q)(q-lambda q)(q^2-lambda q) cdots (q^beta -lambda q) times (1-lambda q^2)(1-lambda q^3)cdots (1-lambda q^{gamma+1})$$
        $$ = prod_{i=0}^beta (q^i-lambda q)times q^{1+2+cdots+gamma}(q^{-1}-lambda q)(q^{-2}-lambda q)cdots(q^{-gamma}-lambda q)
        = q^{1+2+cdots+gamma}prod_{i=-gamma}^beta (q^i-lambda q)
        = q^{frac{gamma(gamma+1)}{2}}prod_{i=-gamma}^beta (q^i-lambda q)$$



        and $Q_Y$ is given by
        $$Q_Y(q) = (1-q)(1-q^2)cdots (1-q^beta) times (1-q^gamma)(1-q^{gamma-1})cdots (1-q) times (1-q^{gamma+beta+1})$$
        Using the notation $[n] = 1-q^n$, $[n]! = [n][n-1]!$ and $[0]!=1$, we can write it as
        $$Q_Y(q) = [beta]![gamma]!(1-q^p)$$
        The remaining piece of information needed to calculate the trace is the trace $text{tr}(pi_Y(b))$ for the Young tableaux $Y = Y_{beta,gamma}$ specified above and $b=(sigma_1sigma_2cdotssigma_{p-1})^m$.



        For coprime $m$ and $p$, this is given by
        $$text{tr}(pi_Y((sigma_1sigma_2cdotssigma_{p-1})^m) = (-1)^gamma q^{beta m}$$



        This is related to the fact that $(sigma_1sigma_2cdotssigma_{p-1})^q$ is in the center of the braid group $B_p$ but I don't know (1) why there is such connection, and (2) why this braid word is in the center.



        Putting the pieces of information together, we have
        $$text{tr}((sigma_1sigma_2cdotssigma_{p-1})^m)
        = sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^gamma q^{beta m}
        left(dfrac{1-q}{1-lambda q} right)^p dfrac{q^{frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q) $$



        Hence the 2-variable Jones polynomial is given by
        $$X_{text{Cl}((sigma_1sigma_2cdotssigma_{p-1})^m)}(q, lambda)
        = left(-dfrac{1-lambda q}{sqrt lambda (1-q)}right)^{p-1} sqrt lambda^{(p-1)m} times
        sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^gamma q^{beta m}
        left(dfrac{1-q}{1-lambda q} right)^p dfrac{q^{frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q)$$

        $$= left(dfrac{1-q}{1-lambda q} right) lambda^{frac{(p-1)(m-1)}{2}}
        sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^{p-1-gamma} q^{beta m}
        dfrac{q^{frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q) $$

        $$= left(dfrac{1-q}{1-lambda q} right) lambda^{frac{(p-1)(m-1)}{2}}
        sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^beta
        dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q) $$





        Chern-Simons Theory approach



        For a general torus link $K(p,m)$, it can be viewed as a $d$-component link, where $d=text{gcd}(p,m)$, with each component being a $(P,M)$-torus knot, where $dfrac{P}{M}$ is the reduced form of the fraction $dfrac{p}{m}$. We can then split the manifold $S^3$ into pieces with toroidal boundaries such that each piece contains one $K(P,M)$, schematically shown below:
        Splitting of the torus link
        Then by the spirit of the partition function, we have
        $$V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{p-1})^m)}(q, lambda) = left(V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{P-1})^M)}(q, lambda) right)^d
        =left( left(dfrac{1-q}{1-lambda q} right) lambda^{frac{(p-d)(m-d)}{2d^2}}
        sum_{beta+gamma+1=frac pd; beta,gamma geq 0} (-1)^beta
        dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^frac pd)}prod_{i=-gamma}^beta (q^i-lambda q) right)^d
        = left(dfrac{1-q}{1-lambda q} right)^d lambda^{frac{(p-d)(m-d)}{2d}}
        left(sum_{beta+gamma+1=frac pd; beta,gamma geq 0} (-1)^beta
        dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^frac pd)}prod_{i=-gamma}^beta (q^i-lambda q) right)^d$$



        Similarly, for the tst links, we have the following splitting of $S^3$ into $(d+1)$ pieces:
        Splitting of the tst link
        where the rightmost knot is the unknot $O$.



        Thus we have
        $$V_{text{Cl}((sigma_1^2 sigma_2 cdots sigma_p)^m)}(q, lambda) = left(V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{P-1})^M)}(q, lambda) right)^d V_{O}(q, lambda)
        = left(V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{P-1})^M)}(q, lambda) right)^d
        = left(dfrac{1-q}{1-lambda q} right)^d lambda^{frac{(p-d)(m-d)}{2d}}
        left(sum_{beta+gamma+1=frac pd; beta,gamma geq 0} (-1)^beta
        dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^frac pd)}prod_{i=-gamma}^beta (q^i-lambda q) right)^d$$





        Ocneanu trace approach



        First consider the torus links. I will start with the easiest case $sigma_1 sigma_2 cdots sigma_p$.
        The Ocneanu trace is obviously $z^n$.



        Then I will try $(sigma_1 sigma_2 cdots sigma_p)^2$. Denote $beta sim_O beta'$ if they have the same Ocneanu traces, i.e. $text{tr}(beta) = text{tr}(beta')$, and $beta sim_M beta'$ if they are related by a sequence of Markov moves. We have
        $$(sigma_1 sigma_2 cdots sigma_p)^2 $$
        $$= sigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_p $$
        $$= sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_p sigma_{p-1} sigma_p $$
        $$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1} sigma_p sigma_{p-1} $$
        $$sim_O z sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}^2 $$
        $$sim_O z(q-1) sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1} + zq sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}$$
        $$sim_O z(q-1) (sigma_1 sigma_2 cdots sigma_{p-1})^2 + z^2q sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-2}$$
        $$sim_O z(q-1) (sigma_1 sigma_2 cdots sigma_{p-1})^2 + z^2q (sigma_1 sigma_2 cdots sigma_{p-2})^2$$
        Writing $tr((sigma_1 sigma_2 cdots sigma_p)^2) = T(2,p)$, we have the recurrence relation
        $$T(2,p) = z(q-1) T(2,p-1) + z^2qT(2,p-2)$$
        which is second-order, homogeneous and has constant coefficients (no dependance on $p$). The characteristic equation is given by
        $$lambda^2 - z(q-1) lambda - z^2q = 0$$
        Solving the quadratic equation gives
        $$lambda = dfrac{z(q-1) pm sqrt{(z(q-1))^2 - 4(-z^2q)}}{2}
        = dfrac{z(q-1) pm sqrt{z^2q^2 - 2z^2q + z^2 + 4z^2q}}{2}
        = dfrac{z(q-1) pm sqrt{z^2(q + 1)^2}}{2}
        = dfrac{z(q-1) pm z(q+1)}{2}$$

        We have $lambda = zq$ or $-z$.
        Hence
        $$T(2,p) = a(zq)^p + b(-z)^p$$ for some $a$ and $b$.



        The initial values are given by
        $$T(2,1) = text{tr}(sigma_1^2) = (q-1)text{tr}(sigma_1) + q = (q-1)z + q$$
        and
        $$T(2,2) = text{tr}(sigma_1sigma_2sigma_1sigma_2) = text{tr}(sigma_1^2sigma_2sigma_1) = ztext{tr}(sigma_1^3) = z(q-1)text{tr}(sigma_1^2) + zq text{tr}(sigma_1) = z(q-1)((q-1)z + q) + z^2q = z^2(q-1)^2 +zq(q-1) + z^2q$$
        Substitution gives
        $$azq - bz = (q-1)z + q$$
        $$a(zq)^2 + bz^2 = z^2(q-1)^2 +zq(q-1) + z^2q$$
        Cramer's rule gives
        $$a =
        dfrac{begin{vmatrix}
        (q-1)z + q & -z \
        z^2(q-1)^2 +zq(q-1) + z^2q & z^2
        end{vmatrix}}
        {begin{vmatrix}
        zq & -z \
        (zq)^2 + zq & z^2
        end{vmatrix}}
        =dfrac{zbegin{vmatrix}
        (q-1)z + q & -1 \
        z^2(q-1)^2 +zq(q-1) + z^2q & z
        end{vmatrix}}
        {z^2qbegin{vmatrix}
        1 & -1 \
        zq + 1 & z
        end{vmatrix}}
        =dfrac{(q-1)z^2 + qz + z^2(q-1)^2 +zq(q-1) + z^2q}{zq(z+zq+1)}
        =dfrac{zq + z^2q(q-1) +zq(q-1) + z^2q}{zq(z+zq+1)}
        =dfrac{q(z + 1)}{z+zq+1}
        $$

        and
        $$b =
        dfrac{begin{vmatrix}
        zq & (q-1)z + q \
        (zq)^2 & z^2(2q-1) + zq
        end{vmatrix}}
        {begin{vmatrix}
        zq & -z \
        (zq)^2 + zq & z^2
        end{vmatrix}}
        =dfrac{zqbegin{vmatrix}
        1 & (q-1)z + q \
        zq & z^2(q-1)^2 +zq(q-1) + z^2q
        end{vmatrix}}{z^2q(z+zq+1)}
        =dfrac{zq (z^2(q-1)^2 +zq(q-1) + z^2q - ((q-1)z^2q + zq^2))}{z^2q(z+zq+1)}
        =dfrac{z(z-q)}{z(z+zq+1)}
        =dfrac{z - q}{z+zq+1}
        $$

        Hence
        $$text{tr}((sigma_1 sigma_2 cdots sigma_p)^2) = dfrac{q(z + 1)(zq)^p}{z+zq+1} + dfrac{(z - q)(-z)^p}{z+zq+1} = dfrac{q(z + 1)(zq)^p + (z - q)(-z)^p}{z+zq+1}$$



        Now we try $(sigma_1 sigma_2 cdots sigma_p)^3$. By the same token, we have
        $$sigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_p$$
        $$=sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_psigma_{p-1}sigma_psigma_1 sigma_2 cdots sigma_p$$
        $$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}sigma_psigma_{p-1}sigma_1 sigma_2 cdots sigma_p$$
        $$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}^2sigma_1 sigma_2 cdots sigma_{p-1}sigma_psigma_{p-1}$$
        $$sim_O zsigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}^2sigma_1 sigma_2 cdots sigma_{p-1}^2$$
        $$sim_O z(q-1)(sigma_1 sigma_2 cdots sigma_{p-1})^3 + zqsigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1}^2$$
        $$sim_O z(q-1)(sigma_1 sigma_2 cdots sigma_{p-1})^3 + zq(q-1)sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1} + zq^2sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-2}$$
        $$sim_O z(q-1)(sigma_1 sigma_2 cdots sigma_{p-1})^3 + zq(q-1)sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1} + z^2q^2(sigma_1 sigma_2 cdots sigma_{p-2})^3$$



        But
        $$sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1} $$
        $$=sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-2}sigma_{p-3}sigma_{p-2} sigma_{p-1}$$
        $$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_{p-3} sigma_{p-1}$$
        $$= sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-1}sigma_{p-2}sigma_{p-1}sigma_{p-3}$$
        $$sim_M sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_{p-1}sigma_{p-2}sigma_{p-3}$$
        $$sim_O zsigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}^2sigma_{p-3}$$
        $$sim_O z ((q-1) sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_{p-3} + qsigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
        $$sim_O z ((q-1) sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-2}sigma_{p-3}sigma_{p-2} + qz(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
        $$sim_O z ((q-1) sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2} + qz(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
        $$= z ((q-1) (sigma_1 sigma_2 cdots sigma_{p-2})^3 + qz(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
        $$sim_O z(q-1) (sigma_1 sigma_2 cdots sigma_{p-2})^3
        + q(q-1)z^2(sigma_1 sigma_2 cdots sigma_{p-3})^3
        +q^2z^2(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-4}$$

        $$sim_M z(q-1) (sigma_1 sigma_2 cdots sigma_{p-2})^3
        + q(q-1)z^2(sigma_1 sigma_2 cdots sigma_{p-3})^3
        +q^2z^2 sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-4} sigma_1 sigma_2 cdots sigma_{p-3}$$

        Writing $text{tr}((sigma_1 sigma_2 cdots sigma_p)^3) = T(3, p)$ and
        $text{tr}(sigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_p) = U(3, p)$, we have
        $$U(3, p-1) = z(q-1)T(3,p-2)+q(q-1)z^2T(3,p-3)+q^2z^2U(3,p-3)$$
        and
        $$T(3,p) = z(q-1)T(3,p-1) + zq(q-1)U(3,p-1) + z^2q^2T(3,p-2)$$
        Rearrange the last relation to give
        $$U(3,p-1) = dfrac{T(3,p)-z(q-1)T(3,p-1)-z^2q^2T(3,p-2)}{zq(q-1)}
        =dfrac{T(3,p)}{zq(q-1)}-dfrac{T(3,p-1)}{q}-dfrac{zqT(3,p-2)}{q-1}$$

        Substitute into the first relation to give
        $$dfrac{T(3,p)}{zq(q-1)}-dfrac{T(3,p-1)}{q}-dfrac{zqT(3,p-2)}{q-1}= z(q-1)T(3,p-2)+q(q-1)z^2T(3,p-3)+q^2z^2left(dfrac{T(3,p-2)}{zq(q-1)}-dfrac{T(3,p-3)}{q}-dfrac{zqT(3,p-4)}{q-1}right)$$



        This recurrence relation needs to be solved subject to
        $$U(3, 2) = text{tr}(sigma_2sigma_1sigma_2) = text{tr}(sigma_1sigma_2sigma_1) = ztext{tr}(sigma_1^2) = z(q-1)text{tr}(sigma_1)+zq = z^2(q-1)+zq$$
        $$T(3, 1) = text{tr}(sigma_1^3) = (q-1)text{tr}(sigma_1^2)+qtext{tr}(sigma_1) = (q-1)^2text{tr}(sigma_1)+ (q-1)q + qz
        = (q-1)^2z + (q-1)q + qz$$

        $$T(3, 2) = text{tr}((sigma_1sigma_2)^3) = text{tr}(sigma_1sigma_2sigma_1sigma_2sigma_1sigma_2) = text{tr}(sigma_2sigma_1sigma_2^2sigma_1sigma_2)
        = (q-1)text{tr}(sigma_2sigma_1sigma_2sigma_1sigma_2) + qtext{tr}(sigma_2sigma_1sigma_1^2sigma_2)
        = (q-1)text{tr}(sigma_2sigma_1sigma_1sigma_2sigma_1) + q(q-1)text{tr}(sigma_2sigma_1sigma_1sigma_2) + q^2text{tr}(sigma_2sigma_1sigma_2)
        = (q-1)^2text{tr}(sigma_2sigma_1sigma_1sigma_2sigma_1) + (q-1)qtext{tr}(sigma_2sigma_2sigma_1)
        + q(q-1)text{tr}(sigma_2^2sigma_1sigma_1) + q^2text{tr}(sigma_1sigma_2sigma_1)
        = (q-1)^2text{tr}(sigma_1sigma_1sigma_2sigma_1sigma_2) + (q-1)qtext{tr}(sigma_2sigma_2sigma_1)
        + q(q-1)((q-1)text{tr}(sigma_2sigma_1^2)+qtext{tr}(sigma_1sigma_1))+ q^2ztext{tr}(sigma_1^2)
        = (q-1)^2text{tr}(sigma_1sigma_1sigma_1sigma_2sigma_1) + (q-1)qtext{tr}(sigma_1sigma_2sigma_1)
        + q(q-1)((q-1)ztext{tr}(sigma_1^2)+qtr(sigma_1^2))+ q^2ztext{tr}(sigma_1^2)
        = (q-1)^2ztext{tr}(sigma_1^4) + (q-1)qztext{tr}(sigma_1^2)
        + q(q-1)((q-1)ztext{tr}(sigma_1^2)+qtr(sigma_1^2))+ q^2ztext{tr}(sigma_1^2)
        = (q-1)^2ztext{tr}(sigma_1^4) + ((q-1)qz + q(q-1)((q-1)z+q)+ q^2z)text{tr}(sigma_1^2)$$

        But
        $$sigma_1^2 sim_O (q-1)sigma_1 + q sim_O (q-1)z + q = qz-z+q$$
        and
        $$sigma_1^4 sim_O (q-1)sigma_1^3 + qsigma_1^2
        sim_O (q-1)((q-1)sigma_1^2 + qsigma_1) + q((q-1)z + q)
        sim_O (q-1)((q-1)((q-1)z + q) + qz) + q((q-1)z + q)$$

        $$=q^3 z - q^2 z + q z - z + q^3 - q^2 + q$$
        Hence
        $$T(3, 2) = (q-1)^2z(q-1)((q-1)((q-1)z + q) + qz) + q((q-1)z + q) + ((q-1)qz + q(q-1)((q-1)z+q)+ q^2z)((q-1)z + q)$$
        $$=q^5 z^2 + q^5 z - 3 q^4 z^2 - 2 q^4 z + q^4 + 6 q^3 z^2 + 4 q^3 z - q^3 - 7 q^2 z^2 - 2 q^2 z + q^2 + 4 q z^2 - z^2$$






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        $endgroup$



        Representation theory approach



        First note that the trace of a braid $b$ from $B_p$ is given by the sum over all irreducible representations of the symmetric group $S_p$ given by the Young tableux $Y$ of $p$ nodes:
        $$text{tr}(b) = sum_Y tilde W_Y(q,lambda) text{tr}(pi_Y(b))$$
        where $tilde W_Y(q,lambda)$ is the weight of the tableau $Y$ and $pi_Y(b)$ is the representation of $b$ corresponding to the tableau $Y$.



        Then it is given that $text{tr}(pi_Y(b)) = 0$ unless $Y$ is of the form with $(beta+1)$ boxes in the first row, then $1$ box in each of the $gamma$ rows that follow:



        $Y_{beta,gamma}=$Young tableau



        The weight of the tableau of this form is given by
        $$tilde W_Y(q,lambda) = left(dfrac{1-q}{1-lambda q} right)^p dfrac{R_Y(q,lambda)}{Q_Y(q)}$$
        where
        $$R_Y(q,lambda) = (1-lambda q)(q-lambda q)(q^2-lambda q) cdots (q^beta -lambda q) times (1-lambda q^2)(1-lambda q^3)cdots (1-lambda q^{gamma+1})$$
        $$ = prod_{i=0}^beta (q^i-lambda q)times q^{1+2+cdots+gamma}(q^{-1}-lambda q)(q^{-2}-lambda q)cdots(q^{-gamma}-lambda q)
        = q^{1+2+cdots+gamma}prod_{i=-gamma}^beta (q^i-lambda q)
        = q^{frac{gamma(gamma+1)}{2}}prod_{i=-gamma}^beta (q^i-lambda q)$$



        and $Q_Y$ is given by
        $$Q_Y(q) = (1-q)(1-q^2)cdots (1-q^beta) times (1-q^gamma)(1-q^{gamma-1})cdots (1-q) times (1-q^{gamma+beta+1})$$
        Using the notation $[n] = 1-q^n$, $[n]! = [n][n-1]!$ and $[0]!=1$, we can write it as
        $$Q_Y(q) = [beta]![gamma]!(1-q^p)$$
        The remaining piece of information needed to calculate the trace is the trace $text{tr}(pi_Y(b))$ for the Young tableaux $Y = Y_{beta,gamma}$ specified above and $b=(sigma_1sigma_2cdotssigma_{p-1})^m$.



        For coprime $m$ and $p$, this is given by
        $$text{tr}(pi_Y((sigma_1sigma_2cdotssigma_{p-1})^m) = (-1)^gamma q^{beta m}$$



        This is related to the fact that $(sigma_1sigma_2cdotssigma_{p-1})^q$ is in the center of the braid group $B_p$ but I don't know (1) why there is such connection, and (2) why this braid word is in the center.



        Putting the pieces of information together, we have
        $$text{tr}((sigma_1sigma_2cdotssigma_{p-1})^m)
        = sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^gamma q^{beta m}
        left(dfrac{1-q}{1-lambda q} right)^p dfrac{q^{frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q) $$



        Hence the 2-variable Jones polynomial is given by
        $$X_{text{Cl}((sigma_1sigma_2cdotssigma_{p-1})^m)}(q, lambda)
        = left(-dfrac{1-lambda q}{sqrt lambda (1-q)}right)^{p-1} sqrt lambda^{(p-1)m} times
        sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^gamma q^{beta m}
        left(dfrac{1-q}{1-lambda q} right)^p dfrac{q^{frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q)$$

        $$= left(dfrac{1-q}{1-lambda q} right) lambda^{frac{(p-1)(m-1)}{2}}
        sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^{p-1-gamma} q^{beta m}
        dfrac{q^{frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q) $$

        $$= left(dfrac{1-q}{1-lambda q} right) lambda^{frac{(p-1)(m-1)}{2}}
        sum_{beta+gamma+1=p; beta,gamma geq 0} (-1)^beta
        dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^p)}prod_{i=-gamma}^beta (q^i-lambda q) $$





        Chern-Simons Theory approach



        For a general torus link $K(p,m)$, it can be viewed as a $d$-component link, where $d=text{gcd}(p,m)$, with each component being a $(P,M)$-torus knot, where $dfrac{P}{M}$ is the reduced form of the fraction $dfrac{p}{m}$. We can then split the manifold $S^3$ into pieces with toroidal boundaries such that each piece contains one $K(P,M)$, schematically shown below:
        Splitting of the torus link
        Then by the spirit of the partition function, we have
        $$V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{p-1})^m)}(q, lambda) = left(V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{P-1})^M)}(q, lambda) right)^d
        =left( left(dfrac{1-q}{1-lambda q} right) lambda^{frac{(p-d)(m-d)}{2d^2}}
        sum_{beta+gamma+1=frac pd; beta,gamma geq 0} (-1)^beta
        dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^frac pd)}prod_{i=-gamma}^beta (q^i-lambda q) right)^d
        = left(dfrac{1-q}{1-lambda q} right)^d lambda^{frac{(p-d)(m-d)}{2d}}
        left(sum_{beta+gamma+1=frac pd; beta,gamma geq 0} (-1)^beta
        dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^frac pd)}prod_{i=-gamma}^beta (q^i-lambda q) right)^d$$



        Similarly, for the tst links, we have the following splitting of $S^3$ into $(d+1)$ pieces:
        Splitting of the tst link
        where the rightmost knot is the unknot $O$.



        Thus we have
        $$V_{text{Cl}((sigma_1^2 sigma_2 cdots sigma_p)^m)}(q, lambda) = left(V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{P-1})^M)}(q, lambda) right)^d V_{O}(q, lambda)
        = left(V_{text{Cl}((sigma_1 sigma_2 cdots sigma_{P-1})^M)}(q, lambda) right)^d
        = left(dfrac{1-q}{1-lambda q} right)^d lambda^{frac{(p-d)(m-d)}{2d}}
        left(sum_{beta+gamma+1=frac pd; beta,gamma geq 0} (-1)^beta
        dfrac{q^{beta m+frac{gamma(gamma+1)}{2}}}{[beta]![gamma]!(1-q^frac pd)}prod_{i=-gamma}^beta (q^i-lambda q) right)^d$$





        Ocneanu trace approach



        First consider the torus links. I will start with the easiest case $sigma_1 sigma_2 cdots sigma_p$.
        The Ocneanu trace is obviously $z^n$.



        Then I will try $(sigma_1 sigma_2 cdots sigma_p)^2$. Denote $beta sim_O beta'$ if they have the same Ocneanu traces, i.e. $text{tr}(beta) = text{tr}(beta')$, and $beta sim_M beta'$ if they are related by a sequence of Markov moves. We have
        $$(sigma_1 sigma_2 cdots sigma_p)^2 $$
        $$= sigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_p $$
        $$= sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_p sigma_{p-1} sigma_p $$
        $$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1} sigma_p sigma_{p-1} $$
        $$sim_O z sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}^2 $$
        $$sim_O z(q-1) sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1} + zq sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}$$
        $$sim_O z(q-1) (sigma_1 sigma_2 cdots sigma_{p-1})^2 + z^2q sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-2}$$
        $$sim_O z(q-1) (sigma_1 sigma_2 cdots sigma_{p-1})^2 + z^2q (sigma_1 sigma_2 cdots sigma_{p-2})^2$$
        Writing $tr((sigma_1 sigma_2 cdots sigma_p)^2) = T(2,p)$, we have the recurrence relation
        $$T(2,p) = z(q-1) T(2,p-1) + z^2qT(2,p-2)$$
        which is second-order, homogeneous and has constant coefficients (no dependance on $p$). The characteristic equation is given by
        $$lambda^2 - z(q-1) lambda - z^2q = 0$$
        Solving the quadratic equation gives
        $$lambda = dfrac{z(q-1) pm sqrt{(z(q-1))^2 - 4(-z^2q)}}{2}
        = dfrac{z(q-1) pm sqrt{z^2q^2 - 2z^2q + z^2 + 4z^2q}}{2}
        = dfrac{z(q-1) pm sqrt{z^2(q + 1)^2}}{2}
        = dfrac{z(q-1) pm z(q+1)}{2}$$

        We have $lambda = zq$ or $-z$.
        Hence
        $$T(2,p) = a(zq)^p + b(-z)^p$$ for some $a$ and $b$.



        The initial values are given by
        $$T(2,1) = text{tr}(sigma_1^2) = (q-1)text{tr}(sigma_1) + q = (q-1)z + q$$
        and
        $$T(2,2) = text{tr}(sigma_1sigma_2sigma_1sigma_2) = text{tr}(sigma_1^2sigma_2sigma_1) = ztext{tr}(sigma_1^3) = z(q-1)text{tr}(sigma_1^2) + zq text{tr}(sigma_1) = z(q-1)((q-1)z + q) + z^2q = z^2(q-1)^2 +zq(q-1) + z^2q$$
        Substitution gives
        $$azq - bz = (q-1)z + q$$
        $$a(zq)^2 + bz^2 = z^2(q-1)^2 +zq(q-1) + z^2q$$
        Cramer's rule gives
        $$a =
        dfrac{begin{vmatrix}
        (q-1)z + q & -z \
        z^2(q-1)^2 +zq(q-1) + z^2q & z^2
        end{vmatrix}}
        {begin{vmatrix}
        zq & -z \
        (zq)^2 + zq & z^2
        end{vmatrix}}
        =dfrac{zbegin{vmatrix}
        (q-1)z + q & -1 \
        z^2(q-1)^2 +zq(q-1) + z^2q & z
        end{vmatrix}}
        {z^2qbegin{vmatrix}
        1 & -1 \
        zq + 1 & z
        end{vmatrix}}
        =dfrac{(q-1)z^2 + qz + z^2(q-1)^2 +zq(q-1) + z^2q}{zq(z+zq+1)}
        =dfrac{zq + z^2q(q-1) +zq(q-1) + z^2q}{zq(z+zq+1)}
        =dfrac{q(z + 1)}{z+zq+1}
        $$

        and
        $$b =
        dfrac{begin{vmatrix}
        zq & (q-1)z + q \
        (zq)^2 & z^2(2q-1) + zq
        end{vmatrix}}
        {begin{vmatrix}
        zq & -z \
        (zq)^2 + zq & z^2
        end{vmatrix}}
        =dfrac{zqbegin{vmatrix}
        1 & (q-1)z + q \
        zq & z^2(q-1)^2 +zq(q-1) + z^2q
        end{vmatrix}}{z^2q(z+zq+1)}
        =dfrac{zq (z^2(q-1)^2 +zq(q-1) + z^2q - ((q-1)z^2q + zq^2))}{z^2q(z+zq+1)}
        =dfrac{z(z-q)}{z(z+zq+1)}
        =dfrac{z - q}{z+zq+1}
        $$

        Hence
        $$text{tr}((sigma_1 sigma_2 cdots sigma_p)^2) = dfrac{q(z + 1)(zq)^p}{z+zq+1} + dfrac{(z - q)(-z)^p}{z+zq+1} = dfrac{q(z + 1)(zq)^p + (z - q)(-z)^p}{z+zq+1}$$



        Now we try $(sigma_1 sigma_2 cdots sigma_p)^3$. By the same token, we have
        $$sigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_p$$
        $$=sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_psigma_{p-1}sigma_psigma_1 sigma_2 cdots sigma_p$$
        $$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}sigma_psigma_{p-1}sigma_1 sigma_2 cdots sigma_p$$
        $$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}^2sigma_1 sigma_2 cdots sigma_{p-1}sigma_psigma_{p-1}$$
        $$sim_O zsigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-1}^2sigma_1 sigma_2 cdots sigma_{p-1}^2$$
        $$sim_O z(q-1)(sigma_1 sigma_2 cdots sigma_{p-1})^3 + zqsigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1}^2$$
        $$sim_O z(q-1)(sigma_1 sigma_2 cdots sigma_{p-1})^3 + zq(q-1)sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1} + zq^2sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-2}$$
        $$sim_O z(q-1)(sigma_1 sigma_2 cdots sigma_{p-1})^3 + zq(q-1)sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1} + z^2q^2(sigma_1 sigma_2 cdots sigma_{p-2})^3$$



        But
        $$sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-1} $$
        $$=sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-2}sigma_{p-3}sigma_{p-2} sigma_{p-1}$$
        $$sim_M sigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_{p-3} sigma_{p-1}$$
        $$= sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-1}sigma_{p-2}sigma_{p-1}sigma_{p-3}$$
        $$sim_M sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_{p-1}sigma_{p-2}sigma_{p-3}$$
        $$sim_O zsigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}^2sigma_{p-3}$$
        $$sim_O z ((q-1) sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_{p-3} + qsigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
        $$sim_O z ((q-1) sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-2}sigma_{p-3}sigma_{p-2} + qz(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
        $$sim_O z ((q-1) sigma_1 sigma_2 cdots sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2}sigma_1 sigma_2 cdots sigma_{p-3}sigma_{p-2} + qz(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
        $$= z ((q-1) (sigma_1 sigma_2 cdots sigma_{p-2})^3 + qz(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-3}^2)$$
        $$sim_O z(q-1) (sigma_1 sigma_2 cdots sigma_{p-2})^3
        + q(q-1)z^2(sigma_1 sigma_2 cdots sigma_{p-3})^3
        +q^2z^2(sigma_1 sigma_2 cdots sigma_{p-3})^2sigma_1 sigma_2 cdots sigma_{p-4}$$

        $$sim_M z(q-1) (sigma_1 sigma_2 cdots sigma_{p-2})^3
        + q(q-1)z^2(sigma_1 sigma_2 cdots sigma_{p-3})^3
        +q^2z^2 sigma_1 sigma_2 cdots sigma_{p-3}sigma_1 sigma_2 cdots sigma_{p-4} sigma_1 sigma_2 cdots sigma_{p-3}$$

        Writing $text{tr}((sigma_1 sigma_2 cdots sigma_p)^3) = T(3, p)$ and
        $text{tr}(sigma_1 sigma_2 cdots sigma_psigma_1 sigma_2 cdots sigma_{p-1}sigma_1 sigma_2 cdots sigma_p) = U(3, p)$, we have
        $$U(3, p-1) = z(q-1)T(3,p-2)+q(q-1)z^2T(3,p-3)+q^2z^2U(3,p-3)$$
        and
        $$T(3,p) = z(q-1)T(3,p-1) + zq(q-1)U(3,p-1) + z^2q^2T(3,p-2)$$
        Rearrange the last relation to give
        $$U(3,p-1) = dfrac{T(3,p)-z(q-1)T(3,p-1)-z^2q^2T(3,p-2)}{zq(q-1)}
        =dfrac{T(3,p)}{zq(q-1)}-dfrac{T(3,p-1)}{q}-dfrac{zqT(3,p-2)}{q-1}$$

        Substitute into the first relation to give
        $$dfrac{T(3,p)}{zq(q-1)}-dfrac{T(3,p-1)}{q}-dfrac{zqT(3,p-2)}{q-1}= z(q-1)T(3,p-2)+q(q-1)z^2T(3,p-3)+q^2z^2left(dfrac{T(3,p-2)}{zq(q-1)}-dfrac{T(3,p-3)}{q}-dfrac{zqT(3,p-4)}{q-1}right)$$



        This recurrence relation needs to be solved subject to
        $$U(3, 2) = text{tr}(sigma_2sigma_1sigma_2) = text{tr}(sigma_1sigma_2sigma_1) = ztext{tr}(sigma_1^2) = z(q-1)text{tr}(sigma_1)+zq = z^2(q-1)+zq$$
        $$T(3, 1) = text{tr}(sigma_1^3) = (q-1)text{tr}(sigma_1^2)+qtext{tr}(sigma_1) = (q-1)^2text{tr}(sigma_1)+ (q-1)q + qz
        = (q-1)^2z + (q-1)q + qz$$

        $$T(3, 2) = text{tr}((sigma_1sigma_2)^3) = text{tr}(sigma_1sigma_2sigma_1sigma_2sigma_1sigma_2) = text{tr}(sigma_2sigma_1sigma_2^2sigma_1sigma_2)
        = (q-1)text{tr}(sigma_2sigma_1sigma_2sigma_1sigma_2) + qtext{tr}(sigma_2sigma_1sigma_1^2sigma_2)
        = (q-1)text{tr}(sigma_2sigma_1sigma_1sigma_2sigma_1) + q(q-1)text{tr}(sigma_2sigma_1sigma_1sigma_2) + q^2text{tr}(sigma_2sigma_1sigma_2)
        = (q-1)^2text{tr}(sigma_2sigma_1sigma_1sigma_2sigma_1) + (q-1)qtext{tr}(sigma_2sigma_2sigma_1)
        + q(q-1)text{tr}(sigma_2^2sigma_1sigma_1) + q^2text{tr}(sigma_1sigma_2sigma_1)
        = (q-1)^2text{tr}(sigma_1sigma_1sigma_2sigma_1sigma_2) + (q-1)qtext{tr}(sigma_2sigma_2sigma_1)
        + q(q-1)((q-1)text{tr}(sigma_2sigma_1^2)+qtext{tr}(sigma_1sigma_1))+ q^2ztext{tr}(sigma_1^2)
        = (q-1)^2text{tr}(sigma_1sigma_1sigma_1sigma_2sigma_1) + (q-1)qtext{tr}(sigma_1sigma_2sigma_1)
        + q(q-1)((q-1)ztext{tr}(sigma_1^2)+qtr(sigma_1^2))+ q^2ztext{tr}(sigma_1^2)
        = (q-1)^2ztext{tr}(sigma_1^4) + (q-1)qztext{tr}(sigma_1^2)
        + q(q-1)((q-1)ztext{tr}(sigma_1^2)+qtr(sigma_1^2))+ q^2ztext{tr}(sigma_1^2)
        = (q-1)^2ztext{tr}(sigma_1^4) + ((q-1)qz + q(q-1)((q-1)z+q)+ q^2z)text{tr}(sigma_1^2)$$

        But
        $$sigma_1^2 sim_O (q-1)sigma_1 + q sim_O (q-1)z + q = qz-z+q$$
        and
        $$sigma_1^4 sim_O (q-1)sigma_1^3 + qsigma_1^2
        sim_O (q-1)((q-1)sigma_1^2 + qsigma_1) + q((q-1)z + q)
        sim_O (q-1)((q-1)((q-1)z + q) + qz) + q((q-1)z + q)$$

        $$=q^3 z - q^2 z + q z - z + q^3 - q^2 + q$$
        Hence
        $$T(3, 2) = (q-1)^2z(q-1)((q-1)((q-1)z + q) + qz) + q((q-1)z + q) + ((q-1)qz + q(q-1)((q-1)z+q)+ q^2z)((q-1)z + q)$$
        $$=q^5 z^2 + q^5 z - 3 q^4 z^2 - 2 q^4 z + q^4 + 6 q^3 z^2 + 4 q^3 z - q^3 - 7 q^2 z^2 - 2 q^2 z + q^2 + 4 q z^2 - z^2$$







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        edited Jan 11 at 11:18

























        answered Dec 3 '18 at 11:31









        wilsonwwilsonw

        467315




        467315






























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