Cocycle Invariants and Oriented Singular Knots

TL;DR

This paper extends quandle cocycle invariants to oriented singular knots using oriented singquandles, providing a new tool that distinguishes certain singular knots and links beyond classical invariants.

Contribution

It introduces oriented singquandles and a new cocycle invariant that captures information about singular knots and links, differentiating them from classical knot invariants.

Findings

Invariant coincides with classical cocycle invariant for classical knots

New invariant distinguishes singular granny knot from singular square knot

Provides additional information about singular knots and links

Abstract

We extend the quandle cocycle invariant to oriented singular knots and links using algebraic structures called \emph{oriented singquandles} and assigning weight functions at both regular and singular crossings. This invariant coincides with the classical cocycle invariant for classical knots but provides extra information about singular knots and links. The new invariant distinguishes the singular granny knot from the singular square knot.