# 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 with $$N$$ crossings with a vector $$\ell=[N,N-1,N-2,.....,2,1,2]$$.

It is possible to write the following closed form

$$J(q) = (-1)^{n_-}q^{n_+-2n_-}{\frac {{q}^{-N+1}+ \left( -1 \right) ^{N}{q}^{N+1}+ \left( -1 \right) ^{N}{q} ^{N+3}+ \left( -1 \right) ^{N}{q}^{N-1} }{{q}^{2}+1}}$$

Such formula works well for the following knots : $$3_1$$, $$5_1$$, $$7_1$$, $$9_1$$, $$K11a367$$, $$K13a4878$$,...; and for the following links : $$L2a1$$, $$L4a1$$, $$L6a3$$, $$L8a14$$,...

My questions are:

1. What is the knot that follows after $$K13a4878$$ in the mentioned sequence? It must be a knot with $$15$$ crossings.

2. What is the link that follows after $$L8a14$$ in the mentioned sequence? It must be a link with $$10$$ crossings.

3. Do you think that the "binomial" formula is correct?

• I still don't see how to interpolate $[N,N-1,\dots 1,2]$. Do you mean $[N,N-1,\dots,2,1]$?
– YCor
Nov 27, 2020 at 14:57
• For example, in the case of trefoil knot, the vector is [3,2,1,2] Nov 27, 2020 at 15:00
• But is the implicit interpolation of the form $[N,N-1,\dots, 2,1,2]$?
– YCor
Nov 27, 2020 at 15:06
• Regardless of the notation, the links in question are the (2,n) torus links (or possibly their mirror images). The knots come from the odd values n=3,5,7,9,... and the links are the even values n=2,4,6,8,.... So the answers to questions 1 and 2 are T(2,15) and T(2,10). Nov 27, 2020 at 15:10
• The answer to question 2 is $L10a118$ ( katlas.org/wiki/L10a118 ) Dec 9, 2020 at 12:58