An Affine Index Polynomial Invariant of Virtual Knots

TL;DR

This paper introduces an affine index polynomial invariant for virtual knots, derived from an affine flat biquandle structure, providing a new tool for distinguishing virtual knots and connecting to Vassiliev invariants.

Contribution

It presents a novel polynomial invariant based on affine biquandle structures, expanding the toolkit for virtual knot classification and relating to existing invariants.

Findings

The invariant distinguishes virtual knots effectively.

It relates to and generalizes previous invariants.

Provides a method to construct Vassiliev invariants from the same data.

Abstract

This paper describes a polynomial invariant of virtual knots that is defined in terms of an integer labeling of the virtual knot diagram. This labeling is seen to derive from an essentially unique structure of affine flat biquandle for flat virtual diagrams. The invariant is discussed in detail with many examples,including its relation to previous invariants of this type and we show how to construct Vassiliev invariants from the same data.