Vestnik Yuzhno-Ural'skogo Gosudarstvennogo Universiteta. Seriya "Matematika. Mekhanika. Fizika"
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Vestnik Yuzhno-Ural'skogo Gosudarstvennogo Universiteta. Seriya "Matematika. Mekhanika. Fizika", 2020, Volume 12, Issue 3, Pages 5–14
DOI: https://doi.org/10.14529/mmph200301
(Mi vyurm451)
 

This article is cited in 2 scientific papers (total in 2 papers)

Mathematics

Tabulation of prime projections of links in the thickened surface of genus 2 with no more than 4 crossings

A. A. Akimova

South Ural State University, Chelyabinsk, Russian Federation
Full-text PDF (878 kB) Citations (2)
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Abstract: In this article, we present the result of the first step of tabulation of prime links in the thickened surface of genus 2 that admit diagrams with no more than 4 crossings. Namely, we describe all three steps of tabulation of prime link projections in the surface of genus 2 with no more than 4 crossings. First, we define primality of a link projection in the surface of genus 2. Second, we tabulate prime link projections in the surface of genus 2 with no more than 4 crossings. For this purpose, it is sufficient to consider graphs having special type and enumerate all possible embeddings of the graphs into the surface of genus 2 giving prime link projections. At this step, we prove some auxiliary statements to simplify enumeration of the embeddings. Finally, we show that all obtained projections are nonequivalent in the sense of homeomorphism of the surface of genus 2 onto itself. Our main result states that there exist exactly 15 pairwise nonequivalent prime link projections in the surface of genus 2 with no more than 4 crossings. Several new and known tricks allow rigorously theoretically prove the completeness of the obtained tabulation, as well as to keep the process within reasonable limits. Further, we intend to use the obtained table to classify prime diagrams, i.e. to obtain table of prime links.
Keywords: prime projection, link, thickened surface of genus 2, tabulation.
Funding agency Grant number
Russian Foundation for Basic Research 20-01-00127
The work was supported by RFBR (grant number 20-01-00127).
Received: 07.07.2020
Document Type: Article
UDC: 515.162.8
Language: English
Citation: A. A. Akimova, “Tabulation of prime projections of links in the thickened surface of genus 2 with no more than 4 crossings”, Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 12:3 (2020), 5–14
Citation in format AMSBIB
\Bibitem{Aki20}
\by A.~A.~Akimova
\paper Tabulation of prime projections of links in the thickened surface of genus 2 with no more than 4 crossings
\jour Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz.
\yr 2020
\vol 12
\issue 3
\pages 5--14
\mathnet{http://mi.mathnet.ru/vyurm451}
\crossref{https://doi.org/10.14529/mmph200301}
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  • https://www.mathnet.ru/eng/vyurm/v12/i3/p5
  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Full-text PDF :44
    References:25
     
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