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 Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 2020, Volume 12, Issue 1, Pages 5–13 (Mi vyurm433)

Mathematics

Classification of prime projections of knots in the thickened torus of genus 2 with at most 4 crossings

A. A. Akimova

South Ural State University, Chelyabinsk, Russian Federation

Abstract: We begin classification of prime knots in the thickened torus of genus 2 having diagrams with at most 4 crossings. To this end, it is enough to construct a table of prime knot projections with at most 4 crossings, and use the table to obtain table of prime diagrams, i. e. table of prime knots. In this paper, we present the result of the first step, i. e. we construct a table of prime projections of knots in the thickened torus of genus 2 having at most 4 crossings. First, we introduce definition of prime projection of a knot in the thickened torus of genus 2. Second, we construct a table of prime projections of knots in the thickened torus of genus 2 having at most 4 crossings. To this end, we enumerate graphs of special type and consider all possible embeddings of the graphs into the torus of genus 2 that lead to prime projections. In order to simplify enumeration of the embeddings, we prove some auxiliary statements. Finally, we prove that all obtained projections are inequivalent. Several known and new tricks allow us to keep the process within reasonable limits and rigorously theoretically prove the completeness of the constructed table.

Keywords: prime projection, knot, thickened torus of genus 2, table.

 Funding Agency Grant Number Russian Foundation for Basic Research 20-01-00127 The reported study was funded by RFBR according to the research project No. 20-01-00127.

DOI: https://doi.org/10.14529/mmph200101

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Citation: A. A. Akimova, “Classification of prime projections of knots in the thickened torus of genus 2 with at most 4 crossings”, Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 12:1 (2020), 5–13

Citation in format AMSBIB
\Bibitem{Aki20} \by A.~A.~Akimova \paper Classification of prime projections of knots in the thickened torus of genus 2 with at most 4 crossings \jour Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz. \yr 2020 \vol 12 \issue 1 \pages 5--13 \mathnet{http://mi.mathnet.ru/vyurm433} \crossref{https://doi.org/10.14529/mmph200101} 

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This publication is cited in the following articles:
1. A. A. Akimova, “Tabulation of prime projections of links in the thickened surface of genus 2 with no more than 4 crossings”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 12:3 (2020), 5–14
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