# Questions tagged [jones-polynomial]

The tag has no usage guidance.

15 questions
Filter by
Sorted by
Tagged with
143 views

### Coloured Jones polynomial at 4th root of unity and Arf invariant

Looking at the link invariants of $\operatorname{SU}(2)$ Chern-Simons theory, if we take the coloured Jones polynomial of a knot K, say $J_N^K$ at fundamental representation $N=2$, then we get the ...
• 81
70 views

### Coloured Jones polynomial of the mirror image of a multicomponent link

This question has been reposted from MathStackExchange It is well understood that the usual Jones polynomial of a knot or link can be related to the Jones polynomial of the mirror image of the knot/...
• 81
243 views

### Are there examples of different knots with identical Jones polynomials and different Seifert Genus?

I had asked this question on math.stackexchange 2 days back but came up empty handed so I wanted to ask it here. Are there known examples of $2$ non equivalent knots that have identical jones ...
• 2,292
1 vote
82 views

### The position of complex points on the kiwi-graph of the Jones polynomial

Consider the "kiwi" graph below (the name came from the resemblance to the bird, the national symbol of New Zealand, the country of V. Jones) i.e., roots of the Jones polynomial for knots (...
• 71
121 views

### Relative strength of Jones and colored Jones polynomials

this is my first post here. I've been studying some Knot Theory and I came to a question concerning invariants. We know that the Jones polynomial is related to the RT-invariant associated to the two-...
1 vote
205 views

### 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 ...
• 165
883 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 ...
• 2,236
1 vote
251 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?
230 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) ...
• 219
230 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 ...
• 1,435
250 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?
• 1,299
361 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 ...
• 1,435
893 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 ...
• 1,435
362 views

### On expressions of colored Jones polynomials

In the paper by Masbaum, it was shown that the colored Jones polynomials for a twist knot $K_p$ can be written as \begin{eqnarray} J_{n}(K_p;q)&=&\sum_{k=0}^{\infty} {\cal C}_{K_p}(k) \...
• 2,273
In his famous paper "Hecke algebra representations of braid groups and link polynomials," (Annals 1987), Jones uses a compatible family of traces $tr_z$ on the Iwahori-Hecke algebras $H(q,n)$ of type \$...