On the Coloring of Pseudoknots

TL;DR

This paper extends classical knot colorability concepts to pseudoknots, introduces a pseudodeterminant, and adapts Conway notation, providing new tools for analyzing pseudoknot properties and colorability.

Contribution

It introduces two new colorability definitions for pseudoknots, defines a pseudodeterminant consistent with classical determinants, and extends Conway notation to pseudoknots.

Findings

Defined pseudodeterminant matching classical determinant for knots

Extended Conway notation to pseudoknots

Provided formulae for pseudodeterminants of pseudoknot families

Abstract

Pseudodiagrams are diagrams of knots where some information about which strand goes over/under at certain crossings may be missing. Pseudoknots are equivalence classes of pseudodiagrams, with equivalence defined by a class of Reidemeister-type moves. In this paper, we introduce two natural extensions of classical knot colorability to this broader class of knot-like objects. We use these definitions to define the determinant of a pseudoknot (i.e. the pseudodeterminant) that agrees with the classical determinant for classical knots. Moreover, we extend Conway notation to pseudoknots to facilitate the investigation of families of pseudoknots and links. The general formulae for pseudodeterminants of pseudoknot families may then be used as a criterion for p-colorability of pseudoknots.