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

We introduce twisted set-theoretic Yang–Baxter solutions and develop an associated cohomology theory, which generalizes the standard cohomology theory of Yang–Baxter solutions. By employing cocycles of twisted biquandles along with Alexander numbering, we construct state-sum invariants for knots and knotted surfaces. As an application, we use our approach to distinguish the [Formula: see text]-twist spun trefoil from its reverse orientation, in line with prior findings.