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

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 → 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.