The Mock Alexander Polynomial for Knotoids and Linkoids

TL;DR

This paper extends the mock Alexander polynomial to knotoids and linkoids, proving a key conjecture and developing new invariants based on this polynomial for these generalized knot-like structures.

Contribution

It proves a conjecture on the mock Alexander polynomial for knotoids and introduces canonical invariants for linkoids derived from this polynomial.

Findings

Proved a conjecture on the mock Alexander polynomial for knotoids.

Generalized the polynomial to linkoids.

Constructed new invariants for linkoids.

Abstract

The mock Alexander polynomial is an extension of the classical Alexander polynomial, defined and studied for (virtual) knots and knotoids by the second and third authors. In this paper we consider the mock Alexander polynomial for generalizations of knotoids. We prove a conjecture on the mock Alexander polynomial for knotoids, which generalizes to uni-linkoids. Afterwards we give constructions for canonical invariants of linkoids derived from the mock Alexander polynomial, using the formalism of generalized knotoids due to Adams et al.