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 ...
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 ...
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?
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) ...
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 ...
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?