Skew Brace Enhancements

TL;DR

This paper introduces new polynomial invariants for virtual knots and links derived from skew brace structures, providing more refined tools for distinguishing these topological objects beyond existing counting invariants.

Contribution

The authors develop two novel polynomial invariants based on skew braces that enhance the biquandle counting invariant for virtual knots and links, demonstrating their effectiveness with examples.

Findings

New single-variable polynomial invariant using skew brace ideals

Two-variable polynomial invariant based on skew brace group structures

Invariants are not determined by the counting invariant, confirming they are proper enhancements

Abstract

We use the structure of skew braces to enhance the biquandle counting invariant for virtual knots and links for finite biquandles defined from skew braces. We introduce two new invariants: a single-variable polynomial using skew brace ideals and a two-variable polynomial using the skew brace group structures. We provide examples to show that the new invariants are not determined by the counting invariant and hence are proper enhancements.