Isotopy and Homotopy Invariants of Classical and Virtual Pseudoknots

TL;DR

This paper develops a Gauss-diagrammatic framework for pseudoknots, introducing new invariants for classical and virtual pseudoknots, and provides tables of unknotting numbers, advancing the classification of these knots and their relation to singular knots.

Contribution

It introduces a novel Gauss-diagrammatic approach to pseudoknots, defining new invariants and classifying homotopy classes, with implications for singular knot theory.

Findings

New invariants for pseudoknots

Tables of unknotting numbers

Connections to singular knot classification

Abstract

Pseudodiagrams are knot or link diagrams where some of the crossing information is missing. Pseudoknots are equivalence classes of pseudodiagrams, where equivalence is generated by a natural set of Reidemeister moves. In this paper, we introduce a Gauss-diagrammatic theory for pseudoknots which gives rise to the notion of a virtual pseudoknot. We provide new, easily computable isotopy and homotopy invariants for classical and virtual pseudodiagrams. We also give tables of unknotting numbers for homotopically trivial pseudoknots and homotopy classes of homotopically nontrivial pseudoknots. Since pseudoknots are closely related to singular knots, this work also has implications for the classification of classical and virtual singular knots.