Universal cocycle Invariants for singular knots and links

TL;DR

This paper introduces a new invariant for singular knots and links using biquandle structures and non-commutative cocycles, generalizing classical invariants and providing computational examples.

Contribution

It develops a universal group and functions for all 2-cocycles in biquandle-based invariants of singular knots, extending previous invariants to non-commutative settings.

Findings

Defined a universal group for 2-cocycles in biquandle invariants.

Provided computational examples including generalizations of linking numbers.

Extended invariants to virtual knots with a self-linking number concept.

Abstract

Given a biquandle $(X, S)$, a function $\tau$ with certain compatibility and a pair of {\em non commutative cocyles} $f,h:X \times X\to G$ with values in a non necessarily commutative group $G$, we give an invariant for singular knots / links. Given $(X,S,\tau)$, we also define a universal group $U_{nc}^{fh}(X)$ and universal functions governing all 2-cocycles in $X$, and exhibit examples of computations. When the target group is abelian, a notion of {\em abelian cocycle pair} is given and the "state sum" is defined for singular knots/links. Computations generalizing linking number for singular knots are given. As for virtual knots, a "self-linking number" may be defined for singular knots