Enhancements of the rack counting invariant via N-reduced dynamical cocycles

TL;DR

This paper introduces N-reduced dynamical cocycles to enhance the rack counting invariant, providing new knot invariants that distinguish knots beyond traditional invariants like the Jones polynomial.

Contribution

The paper develops N-reduced dynamical cocycles and demonstrates their effectiveness in creating enhanced invariants for classical and virtual knots.

Findings

New invariants are not determined by existing invariants

Examples show enhanced discriminative power

Invariants apply to both classical and virtual knots

Abstract

We introduce the notion of N-reduced dynamical cocycles and use these objects to define enhancements of the rack counting invariant for classical and virtual knots and links. We provide examples to show that the new invariants are not determined by the rack counting invariant, the Jones polynomial or the generalized Alexander polynomial.