Virtual link and knot invariants from non-abelian Yang-Baxter 2-cocycle pairs

TL;DR

This paper introduces a new invariant for virtual and classical knots/links derived from non-abelian Yang-Baxter 2-cocycle pairs, expanding previous invariants with a broader algebraic framework and computational tools.

Contribution

It defines a novel invariant using non-commutative 2-cocycles pairs for set-theoretic solutions of Yang-Baxter, including a related group and computational methods.

Findings

Defined a new knot/link invariant from non-abelian 2-cocycles

Constructed a group U_{nc}^{fg} governing 2-cocycles

Performed example computations using GAP

Abstract

For a given $(X,S,\beta)$, where $S,\beta\colon X\times X\to X\times X$ are set theoretical solutions of Yang-Baxter equation with a compatibility condition, we define an invariant for virtual (or classical) knots/links using non commutative 2-cocycles pairs $(f,g)$ that generalizes the one defined in [FG2]. We also define, a group $U_{nc}^{fg}=U_{nc}^{fg}(X,S,\beta)$ and functions $\pi_f, \pi_g\colon X\times X\to U_{nc}^{fg}(X)$ governing all 2-cocycles in $X$. We exhibit examples of computations achieved using GAP.