A state sum invariant for regular isotopy of links having a polynomial number of states

TL;DR

This paper introduces the VSE-invariant, a new regular isotopy link invariant that generalizes the Jones Polynomial, offering a computationally efficient alternative with fewer states for distinguishing links, especially with many crossings.

Contribution

The paper presents the VSE-invariant, a stronger and more computationally feasible link invariant than Jones', with specializations that reduce the number of states needed for calculations.

Findings

VSE-invariant distinguishes links that Jones' polynomial cannot.

Specializations of VSE-invariant have polynomially many states, e.g., O(n^k).

Efficient computation for links with many crossings, e.g., n=500.

Abstract

The state sum regular isotopy invariant of links which I introduce in this work is a generalization of the Jones Polynomial. So it distinguishes any pair of links which are distinguishable by Jones'. This new invariant, denoted {\em VSE-invariant} is strictly stronger than Jones': I detected a pair of links which are not distinguished by Jones' but are distinguished by the new invariant. The full VSE-invariant has $3^n$ states. However, there are useful specializations of it parametrized by an integer k, having $O(n^k)=\sum_{\ell=0}^k {n \choose \ell} 2^\ell $ states. The link with more crossings of the pair which was distinguished by the VSE-invariant has 20 crossings. The specialization which is enough to distinguish corresponds to k=2 and has only 801 states, as opposed to the $2^{20} = 1,048,576$ states of the Jones polynomial of the same link. The full VSE-invariant of it has $3^{20} = 3,486,784,401$ states. The VSE-invariant is a good alternative for the Jones polynomial when the number of crossings makes the computation of this polynomial impossible. For instance, for $k=2$ the specialization of the VSE-invariant of a link with $n=500$ crossings can be computed in a few minutes, since it has only $2 n^2+1 = 500,001$ states.