Singquandle Shadows and Singular knot Invariants

TL;DR

This paper introduces shadow structures for singular knot theory, defining two new invariants that generalize classical quandle colorings and include a polynomial invariant, with explicit examples provided.

Contribution

It develops the concept of shadow structures for singular knots, creating new invariants that extend classical quandle-based invariants to singular knot theory.

Findings

Defined a shadow counting invariant for singular links

Introduced a shadow polynomial invariant for shadow structures

Enhanced invariants by combining counting and polynomial invariants

Abstract

We introduce shadow structures for singular knot theory. Precisely, we define \emph{two} invariants of singular knots and links. First, we introduce a notion of action of a singquandle on a set to define a shadow counting invariant of singular links which generalize the classical shadow colorings of knots by quandles. We then define a shadow polynomial invariant for shadow structures. Lastly, we enhance the shadow counting invariant by combining both the shadow counting invariant and the shadow polynomial invariant. Explicit examples of computations are given.