The Jones and Alexander polynomials for singular links

TL;DR

This paper develops extended state models for Jones and Alexander polynomials applicable to singular links and long singular knots, enabling the detection of non-invertibility and introducing multi-variable polynomial invariants.

Contribution

It introduces new multi-variable polynomial invariants for singular links and long singular knots, extending classical polynomials to handle singularities and multiple variables.

Findings

Extended state models for Jones and Alexander polynomials to singular links.

Polynomials with d+1 variables for long singular knots with d double points.

Ability to detect non-invertibility of long singular knots.

Abstract

We extend the state models for Jones and Alexander polynomials of classical links to state models of 2-variable polynomials in the case of singular links. Moreover, we extend both of them to polynomials with d+1 variables for long singular knots with exactly d double points. These extensions can detect non-invertibility of long singular knots.