Questions tagged [jones-polynomial]

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Possible “binomial” formula for the Jones polynomial

The following conjectural "binomial" formula for the Jones polynomials $$J(q)=(-1)^{n_-}q^{n_+-2n_-}\left(\sum_{k=0}^N\binom{N}{k}(-q)^k (q+1/q)^{\ell_{k+1}-1}\right)$$ is for a knot or link ...
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2answers
368 views

What are applications of Jones polynomial on von Neumann algebras?

I have read according list of below papers a basic connection between Jones polynomial and statistical mechanics is that the Kauffman bracket or Kauffman polynomial a polynomial invariant of knots is ...
1
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1answer
119 views

Jones polynomial of cable knots

Let $K_{p,q}$ be a $(p,q)$-cable of the non-trivial knot $K$ in $S^3$. Is there a closed formula for the Jones polynomial for $K_{p,q}$ as in the case of Alexander polynomial or Seifert matrices?
2
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1answer
160 views

Easy lemma for trivalent graphs in colored Jones polynomial

In his 2008 paper, Tanaka, Toshifumi, The colored Jones polynomials of doubles of knots, J. Knot Theory Ramifications 17, No. 8, 925-937 (2008). ZBL1149.57023. Tanaka stated a lemma (Lemma 3.3) ...
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1answer
151 views

Proving knot polynomial dependencies and skein relations

I have two questions: From the definition of the Jones polynomial as the normalization of the Kauffman bracket $(-A^3)^{-w(D)} \langle D\rangle$ and substituting $A\rightarrow t^{-1/4}$, how does one ...
8
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1answer
171 views

Set of Jones polynomials as the knot varies

Is a characterization known for the set of Laurent polynomials arising as the Jones polynomial of some knot? More generally, is such a characterization known for any of the famous knot polynomials?
5
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1answer
236 views

Categorifying skein algebras?

We can obtain the Jones polynomial by the Temperly-Lieb algebra and the HOMFLYPT polynomial from the Hecke algebra. Were there attempts to categorify the algebras itself and obtain the Khovanov ...
6
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1answer
455 views

HOMFLYPT vs. Jones vs. Alexander polynomial?

I'm searching for examples (perhaps the simplest one?) to show that the HOMFLYPT polynomial is stronger than the Jones and Alexander polynomial, respectively. Any ideas what is the 1st knot in the ...