The Theory of Pseudoknots

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 m...

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Bibliographic Details
Published in:arXiv.org 2012-10
Main Authors: Henrich, Allison, Hoberg, Rebecca, Slavik Jablan, Johnson, Lee, Minten, Elizabeth, Radovic, Ljiljana
Format: Article
Language:English
Subjects:
Chirality
Knots
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1210.6934