The Theory of Pseudoknots

TL;DR

This paper introduces the concept of pseudoknots as equivalence classes of pseudodiagrams with missing crossing information, and proposes the weighted resolution set as an invariant for their classification.

Contribution

It defines pseudoknots, introduces the weighted resolution set invariant, and explores their properties and classifications.

Findings

Weighted resolution set computed for several pseudoknot families

Discussion of crossing number, homotopy, and chirality for pseudoknots

Establishment of a framework for classifying pseudoknots

Abstract

Classical knots in $\mathbb{R}^3$ can be represented by diagrams in the plane. These diagrams are formed by curves with a finite number of transverse crossings, where each crossing is decorated to indicate which strand of the knot passes over at that point. A pseudodiagram is a knot diagram that may be missing crossing information at some of its crossings. At these crossings, it is undetermined which strand passes over. Pseudodiagrams were first introduced by Ryo Hanaki in 2010. Here, we introduce the notion of a pseudoknot, i.e. an equivalence class of pseudodiagrams under an appropriate choice of Reidemeister moves. In order to begin a classification of pseudoknots, we introduce the concept of a weighted resolution set, an invariant of pseudoknots. We compute the weighted resolution set for several pseudoknot families and discuss notions of crossing number, homotopy, and chirality for pseudoknots.