Finding the Alexander polynomial of the following braid closure
$begingroup$
How do you find the Alexander polynomial of the closure of the following braid,
$(sigma_1^{-2}sigma_2^{-1}sigma_3^{-1}sigma_4^{-1}sigma_5^{-1}...sigma_{A-1}^{-1})^B$ where $A$ and $B$ are positive integers?
I have found the general form of the Seifert matrix of link, and then tried to use the formula
$$Delta(t) = det (V-tV^T) $$
where $V$ is the Seifert matrix and $Delta(t)$ is the Alexander polynomial.
I therefore tried using MATLAB to calculate the Alexander polynomials of the knots for $A$ and $B$ running from $1$ to $10$. From the results I guessed the general form of the Alexander polynomial, but I do not end up in a rigorous proof.
I am not quite familiar with the computation of Alexander polynomials using other techniques such as the knot group and Jacobian, so I am not sure if those routes would give a good way in attacking the problem.
PS. I did calculate the Jones polynomial quite quickly, but I don't see any help from that.
knot-theory
asked Jun 4 '13 at 6:17
wilsonwwilsonw
467315
$endgroup$
add a comment |
$begingroup$
How do you find the Alexander polynomial of the closure of the following braid,
$(sigma_1^{-2}sigma_2^{-1}sigma_3^{-1}sigma_4^{-1}sigma_5^{-1}...sigma_{A-1}^{-1})^B$ where $A$ and $B$ are positive integers?
I have found the general form of the Seifert matrix of link, and then tried to use the formula
$$Delta(t) = det (V-tV^T) $$
where $V$ is the Seifert matrix and $Delta(t)$ is the Alexander polynomial.
I therefore tried using MATLAB to calculate the Alexander polynomials of the knots for $A$ and $B$ running from $1$ to $10$. From the results I guessed the general form of the Alexander polynomial, but I do not end up in a rigorous proof.
I am not quite familiar with the computation of Alexander polynomials using other techniques such as the knot group and Jacobian, so I am not sure if those routes would give a good way in attacking the problem.
PS. I did calculate the Jones polynomial quite quickly, but I don't see any help from that.
knot-theory
asked Jun 4 '13 at 6:17
wilsonwwilsonw
467315
$endgroup$
add a comment |
$begingroup$
How do you find the Alexander polynomial of the closure of the following braid,
$(sigma_1^{-2}sigma_2^{-1}sigma_3^{-1}sigma_4^{-1}sigma_5^{-1}...sigma_{A-1}^{-1})^B$ where $A$ and $B$ are positive integers?
I have found the general form of the Seifert matrix of link, and then tried to use the formula
$$Delta(t) = det (V-tV^T) $$
where $V$ is the Seifert matrix and $Delta(t)$ is the Alexander polynomial.
I therefore tried using MATLAB to calculate the Alexander polynomials of the knots for $A$ and $B$ running from $1$ to $10$. From the results I guessed the general form of the Alexander polynomial, but I do not end up in a rigorous proof.
I am not quite familiar with the computation of Alexander polynomials using other techniques such as the knot group and Jacobian, so I am not sure if those routes would give a good way in attacking the problem.
PS. I did calculate the Jones polynomial quite quickly, but I don't see any help from that.
knot-theory
asked Jun 4 '13 at 6:17
wilsonwwilsonw
467315
$endgroup$
How do you find the Alexander polynomial of the closure of the following braid,
$(sigma_1^{-2}sigma_2^{-1}sigma_3^{-1}sigma_4^{-1}sigma_5^{-1}...sigma_{A-1}^{-1})^B$ where $A$ and $B$ are positive integers?
I have found the general form of the Seifert matrix of link, and then tried to use the formula
$$Delta(t) = det (V-tV^T) $$
where $V$ is the Seifert matrix and $Delta(t)$ is the Alexander polynomial.
I therefore tried using MATLAB to calculate the Alexander polynomials of the knots for $A$ and $B$ running from $1$ to $10$. From the results I guessed the general form of the Alexander polynomial, but I do not end up in a rigorous proof.
I am not quite familiar with the computation of Alexander polynomials using other techniques such as the knot group and Jacobian, so I am not sure if those routes would give a good way in attacking the problem.
PS. I did calculate the Jones polynomial quite quickly, but I don't see any help from that.
knot-theory
knot-theory
asked Jun 4 '13 at 6:17
wilsonwwilsonw
467315
asked Jun 4 '13 at 6:17
wilsonwwilsonw
467315
edited Jun 4 '13 at 18:25
wilsonw
asked Jun 4 '13 at 6:17
wilsonwwilsonw
467315
asked Jun 4 '13 at 6:17
wilsonwwilsonw
467315
asked Jun 4 '13 at 6:17
wilsonwwilsonw
467315
467315
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The braid you mention is an element of $B_A$ I assume? It doesn't really matter if you add strings after this, as taking the braid closure will just add a new unlink for every new string.
In any case, it seems like you would be interested in the Burau representation of the braid group. The Alexander polynomial of a braid closure $hat{beta}$ is given by the determinant of the matrix $I-beta_*$ where $beta_*$ is the reduced Burau representation of the braid $beta$. There is a large amount of literature on the Burau representation of a braid going back to the 60s and 70s. I would strongly suggest picking up Birman's book1 for a good introduction to theory behind the representations of the braid groups and how to calculate such maps.
1J. S. Birman Braids, Links and Mapping Class Groups, Annals of Mathematical Studies 82, Princeton University Press, 1975.
edited Dec 3 '18 at 11:59
amWhy
1
answered Jun 4 '13 at 12:47
Dan RustDan Rust
22.7k114884
$endgroup$
$begingroup$
I read your reference book, and tried to get to the answer using the theory of mapping class groups, but I still get to the problem: I have to find the determinant of a matrix, but it is so complicated and it is hard to resolve it.
$endgroup$
– wilsonw
Jun 15 '13 at 6:22
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f410782%2ffinding-the-alexander-polynomial-of-the-following-braid-closure%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The braid you mention is an element of $B_A$ I assume? It doesn't really matter if you add strings after this, as taking the braid closure will just add a new unlink for every new string.
In any case, it seems like you would be interested in the Burau representation of the braid group. The Alexander polynomial of a braid closure $hat{beta}$ is given by the determinant of the matrix $I-beta_*$ where $beta_*$ is the reduced Burau representation of the braid $beta$. There is a large amount of literature on the Burau representation of a braid going back to the 60s and 70s. I would strongly suggest picking up Birman's book1 for a good introduction to theory behind the representations of the braid groups and how to calculate such maps.
1J. S. Birman Braids, Links and Mapping Class Groups, Annals of Mathematical Studies 82, Princeton University Press, 1975.
edited Dec 3 '18 at 11:59
amWhy
1
answered Jun 4 '13 at 12:47
Dan RustDan Rust
22.7k114884
$endgroup$
$begingroup$
I read your reference book, and tried to get to the answer using the theory of mapping class groups, but I still get to the problem: I have to find the determinant of a matrix, but it is so complicated and it is hard to resolve it.
$endgroup$
– wilsonw
Jun 15 '13 at 6:22
add a comment |
$begingroup$
The braid you mention is an element of $B_A$ I assume? It doesn't really matter if you add strings after this, as taking the braid closure will just add a new unlink for every new string.
In any case, it seems like you would be interested in the Burau representation of the braid group. The Alexander polynomial of a braid closure $hat{beta}$ is given by the determinant of the matrix $I-beta_*$ where $beta_*$ is the reduced Burau representation of the braid $beta$. There is a large amount of literature on the Burau representation of a braid going back to the 60s and 70s. I would strongly suggest picking up Birman's book1 for a good introduction to theory behind the representations of the braid groups and how to calculate such maps.
1J. S. Birman Braids, Links and Mapping Class Groups, Annals of Mathematical Studies 82, Princeton University Press, 1975.
edited Dec 3 '18 at 11:59
amWhy
1
answered Jun 4 '13 at 12:47
Dan RustDan Rust
22.7k114884
$endgroup$
$begingroup$
I read your reference book, and tried to get to the answer using the theory of mapping class groups, but I still get to the problem: I have to find the determinant of a matrix, but it is so complicated and it is hard to resolve it.
$endgroup$
– wilsonw
Jun 15 '13 at 6:22
add a comment |
$begingroup$
The braid you mention is an element of $B_A$ I assume? It doesn't really matter if you add strings after this, as taking the braid closure will just add a new unlink for every new string.
In any case, it seems like you would be interested in the Burau representation of the braid group. The Alexander polynomial of a braid closure $hat{beta}$ is given by the determinant of the matrix $I-beta_*$ where $beta_*$ is the reduced Burau representation of the braid $beta$. There is a large amount of literature on the Burau representation of a braid going back to the 60s and 70s. I would strongly suggest picking up Birman's book1 for a good introduction to theory behind the representations of the braid groups and how to calculate such maps.
1J. S. Birman Braids, Links and Mapping Class Groups, Annals of Mathematical Studies 82, Princeton University Press, 1975.
edited Dec 3 '18 at 11:59
amWhy
1
answered Jun 4 '13 at 12:47
Dan RustDan Rust
22.7k114884
$endgroup$
The braid you mention is an element of $B_A$ I assume? It doesn't really matter if you add strings after this, as taking the braid closure will just add a new unlink for every new string.
In any case, it seems like you would be interested in the Burau representation of the braid group. The Alexander polynomial of a braid closure $hat{beta}$ is given by the determinant of the matrix $I-beta_*$ where $beta_*$ is the reduced Burau representation of the braid $beta$. There is a large amount of literature on the Burau representation of a braid going back to the 60s and 70s. I would strongly suggest picking up Birman's book1 for a good introduction to theory behind the representations of the braid groups and how to calculate such maps.
1J. S. Birman Braids, Links and Mapping Class Groups, Annals of Mathematical Studies 82, Princeton University Press, 1975.
edited Dec 3 '18 at 11:59
amWhy
1
answered Jun 4 '13 at 12:47
Dan RustDan Rust
22.7k114884
edited Dec 3 '18 at 11:59
amWhy
1
edited Dec 3 '18 at 11:59
amWhy
1
edited Dec 3 '18 at 11:59
amWhy
1
1
answered Jun 4 '13 at 12:47
Dan RustDan Rust
22.7k114884
answered Jun 4 '13 at 12:47
Dan RustDan Rust
22.7k114884
answered Jun 4 '13 at 12:47
Dan RustDan Rust
22.7k114884
22.7k114884
$begingroup$
I read your reference book, and tried to get to the answer using the theory of mapping class groups, but I still get to the problem: I have to find the determinant of a matrix, but it is so complicated and it is hard to resolve it.
$endgroup$
– wilsonw
Jun 15 '13 at 6:22
add a comment |
$begingroup$
I read your reference book, and tried to get to the answer using the theory of mapping class groups, but I still get to the problem: I have to find the determinant of a matrix, but it is so complicated and it is hard to resolve it.
$endgroup$
– wilsonw
Jun 15 '13 at 6:22
$begingroup$
I read your reference book, and tried to get to the answer using the theory of mapping class groups, but I still get to the problem: I have to find the determinant of a matrix, but it is so complicated and it is hard to resolve it.
$endgroup$
– wilsonw
Jun 15 '13 at 6:22
$begingroup$
I read your reference book, and tried to get to the answer using the theory of mapping class groups, but I still get to the problem: I have to find the determinant of a matrix, but it is so complicated and it is hard to resolve it.
$endgroup$
– wilsonw
Jun 15 '13 at 6:22
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f410782%2ffinding-the-alexander-polynomial-of-the-following-braid-closure%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
