7
votes
Accepted
Set of Jones polynomials as the knot varies
I believe your question is open for the Jones polynomial. However, it is solved for the Alexander polynomial. On page 171 of Rolfsen's Knots and Links, the following theorem appears.
Theorem. Let $p(...
- 306
7
votes
HOMFLYPT vs. Jones vs. Alexander polynomial?
I will give an example that is likely not the simplest one. The example comes from the paper Behavior of knot invariants under genus 2 mutation by Dunfield, Garoufalidis, Shumakovitch, and ...
- 306
5
votes
Accepted
What are applications of Jones polynomial on von Neumann algebras?
I don't think it's quite right to think of knot polynomials as having applications to von Neumann algebras. Instead I think it's more accurate to say that the Temperley-Lieb-Jones algebras (and more ...
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4
votes
What are applications of Jones polynomial on von Neumann algebras?
In this paper, Vaughan attributes the observation of the similarity of the Temperley-Lieb relations and the braid group relations to D. Hatt, P. de la Harpe and N. Stoltzfus:
Jones, Vaughan, Groupes ...
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3
votes
Accepted
Easy lemma for trivalent graphs in colored Jones polynomial
Rewrite each edge of the graph, labeled by $k$, as $k$ strands with the $k$-th JW idempotent in the middle. Make a similar modification at the vertices. Expand the sums appearing to one side of the $...
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3
votes
Traces on Hecke algebras and the Jones polynomial
This is a beautiful question that is not trivial at all. After Jones' paper, Lambropoulou constructed a trace on the generalized Hecke algebra of type B, through which, she obtained the analogue of ...
- 31
2
votes
Accepted
Categorifying skein algebras?
Much of the research in knot homology has been about categorifying these algebras!
Khovanov's paper math/0103190 is devoted to defining and studying the Temperley-Lieb 2-category which is a ...
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2
votes
Proving knot polynomial dependencies and skein relations
For 1, see Lickorish's "An Introduction to Knot Theory" Proposition 3.7 (page 28).
For 2, the substitution $l=i\alpha$ and $m=-i z$ gives the $\alpha P(L_+)-\alpha^{-1}P(L_-)=z P(L_0)$ version of the ...
- 413
1
vote
Accepted
Jones polynomial of cable knots
As Ian Agol mentioned, if there were a closed formula for the Jones polynomial $V(K_{p,q})$ in terms of $V(K)$, this would give a closed formula for the colored Jones polynomials $V_n(K)$ in terms of ...
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