A Linking Number Definition of the Affine Index Polynomial and Applications

TL;DR

This paper introduces a new linking number-based definition of the Affine Index Polynomial, explores its applications in virtual knot theory, and demonstrates its ability to detect certain knot mutations and distinguish mutant knots.

Contribution

It provides an alternative definition of the Affine Index Polynomial using virtual linking numbers and applies it to solve conjectures and detect mutations in virtual knots.

Findings

Proves the Cosmetic Crossing Change Conjecture for odd and pure virtual knots.

Shows the polynomial detects mutations by positive rotation.

Demonstrates the polynomial cannot detect mutations by positive reflection.

Abstract

This paper gives an alternate definition of the Affine Index Polynomial (called the Wriggle Polynomial) using virtual linking numbers and explores applications of this polynomial. In particular, it proves the Cosmetic Crossing Change Conjecture for odd virtual knots and pure virtual knots. It also demonstrates that the polynomial can detect mutations by positive rotation and proves it cannot detect mutations by positive reflection. Finally it exhibits a pair of mutant knots that can be distinguished by a Type 2 Vassiliev Invariant coming from the polynomial.