A new two-variable generalization of the Jones polynomial

TL;DR

This paper introduces a novel two-variable generalization of the Jones polynomial, enhancing its ability to distinguish non-isotopic links, with rigorous proofs of its well-definedness and a combinatorial formula.

Contribution

It presents a new 2-variable link invariant extending the Jones polynomial, with proofs of its consistency and a combinatorial expression.

Findings

Can distinguish more non-isotopic links than the original Jones polynomial

Provides algebraic and diagrammatic proofs of well-definedness

Includes a closed combinatorial formula for the new invariant

Abstract

We present a new 2-variable generalization of the Jones polynomial that can be defined through the skein relation of the Jones polynomial. The well-definedness of this new generalization is proved both algebraically and diagrammatically as well as via a closed combinatorial formula. This new invariant is able to distinguish more pairs of non-isotopic links than the original Jones polynomial such as the Thistlethwaite link from the unlink with two components.