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?

  • 2
    $\begingroup$ I still don't see how to interpolate $[N,N-1,\dots 1,2]$. Do you mean $[N,N-1,\dots,2,1]$? $\endgroup$
    – YCor
    Nov 27, 2020 at 14:57
  • $\begingroup$ For example, in the case of trefoil knot, the vector is [3,2,1,2] $\endgroup$ Nov 27, 2020 at 15:00
  • $\begingroup$ But is the implicit interpolation of the form $[N,N-1,\dots, 2,1,2]$? $\endgroup$
    – YCor
    Nov 27, 2020 at 15:06
  • 2
    $\begingroup$ 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). $\endgroup$ Nov 27, 2020 at 15:10
  • 1
    $\begingroup$ The answer to question 2 is $L10a118$ ( katlas.org/wiki/L10a118 ) $\endgroup$ Dec 9, 2020 at 12:58


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