Twisted Yang-Baxter sets, cohomology theory, and application to knots

Mohamed Elhamdadi, Manpreet Singh

Published 2024-01-03Version 1

In this article, we introduce a notion of twisted set-theoretic Yang-Baxter solution, which is a triplet $(X,f,R)$, where $(X,R)$ is a Yang-Baxter set and $f:X \to X$ is an automorphism of $(X,R)$. We present a cohomology theory for it, and use cocycles of twisted biquandles in amalgamation with Alexander numbering to construct state-sum invariant of knots and knotted surfaces. Additionally, we introduce a twisted version of cohomology theory for Yang-Baxter sets and give applications to knot theory.