CWR sequence of invariants of alternating links and its properties

TL;DR

The paper introduces the CWR invariant, a new polynomial-based invariant for alternating links that generalizes previous invariants and offers stronger distinguishing power, supported by recursive relations and formulas derived from graph theory.

Contribution

It presents the CWR invariant, extending the WRP invariant with enhanced strength and explicit formulas, along with recursive skein relations for alternating links.

Findings

CWR invariant is stronger than WRP, HOMFLYPT, and Kauffman polynomials.

Derived recursive skein relations for the CWR invariant.

Provided explicit formulas using weighted adjacency matrices of Tait graphs.

Abstract

We present the $CWR$ invariant, a new invariant for alternating links, which builds upon and generalizes the $WRP$ invariant. The $CWR$ invariant is an array of two-variable polynomials that provides a stronger invariant compared to the $WRP$ invariant. We compare the strength of our invariant with the classical HOMFLYPT, Kauffman $3$-variable, and Kauffman $2$-variable polynomials on specific knot examples. Additionally, we derive general recursive "skein" relations, and also specific formulas for the initial components of the $CWR$ invariant using weighted adjacency matrices of modified Tait graphs.