Classification of Links of Small Complexity in the Thickened Torus

Akimova, A. A.; Matveev, Sergei; Tarkaev, V. V.

doi:10.1134/s008154381809002x

Cited by 8 publications

(4 citation statements)

References 12 publications

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“…Indeed, not essential projections correspond to links that can be found in the classifications of links in the 3-dimensional sphere S 3 [1][2][3], solid torus (thickened annulus A×I) or thickened torus T×I [12]. Here we note that today there exist no classification of links in the solid torus, but, as well as in the case of knots, we consider the construction of such a classification as an independent problem, which is beyond the scope of our interests in this article.…”

Section: (A)mentioning

confidence: 99%

“…In this article, in order to obtain all the projections, we use the method of removing not transversal points [12] and the cutting technique [15].…”

Section: Lemma 2 (Lemma 3 In [15]mentioning

confidence: 99%

“…As regards tabulation of global links, note classifications of links in the projective space [11] and prime links in the thickened torus [12]. Also, note a classification of virtual links of special type, namely, alternating virtual links [13], see also [14] for the associated database, which include alternating virtual knots as well.…”

Section: Introductionmentioning

confidence: 99%

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Tabulation of Prime Projections of Links in the Thickened Surface of Genus 2 with no more than 4 Crossings

Akimova¹

2020

MMPh

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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.

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Section: (A)mentioning

confidence: 99%

“…In this article, in order to obtain all the projections, we use the method of removing not transversal points [12] and the cutting technique [15].…”

Section: Lemma 2 (Lemma 3 In [15]mentioning

confidence: 99%

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Tabulation of Prime Projections of Links in the Thickened Surface of Genus 2 with no more than 4 Crossings

Akimova¹

2020

MMPh

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“…Â êëàññè÷åñêîì ñëó÷àå çàöåïëåíèÿ â S 3 çàäàþòñÿ ñ ïîìîùüþ ïðîåêöèé íà ñôåðó S 2 . Â ñëó÷àå çàöåïëåíèé â óòîëù¼ííîì òîðå çàöåïëåíèÿ çàäàþòñÿ ïðîåêöèåé íà òîð [4,5]. Â äàííîé ðàáîòå áóäåì ïðåäñòàâëÿòü çàöåïëåíèÿ â óòîë-ù¼ííîé áóòûëêå Êëåéíà ñ ïîìîùüþ ïðîåêöèé íà áóòûëêó Êëåéíà, êàê è â ðàáîòàõ [6,7].…”

Section: ïðåäñòàâëåíèÿ çàöåïëåíèé â óòîëù¼ííîé áóòûëêå êëåéíàunclassified

Classification of links in the thickened Klein bottle

Nabeeva¹

2020

Sib. Elektron. Mat. Izv.

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Mathematical modelling of biology processes based on the table of prime links in the solid torus up to 4 crossings

Akimova

2021

J. Phys.: Conf. Ser.

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The motivation to study links in lens spaces can be justified in recent applications to theoretical physics and biology. In this paper, we give a short outline of practical significance and implementation proposals of such links, the study of which begins with the study of links in the solid torus, since a punctured disk diagram of a link in the lens space can be considered as a punctured disk diagram in the solid torus provided with the additional slide move. However, links in the solid torus find their applications themselves as well. In this paper, we propose a method to represent knotted proteins as links in the solid torus. Such a method is based on the existence of correspondence between knotoids and knots in the solid torus using a double branched cover. To this end, the table of links in solid torus is necessary. Therefore, we classify all prime links in the solid torus up to 4 crossings. One of possible future applications of the constructed table is an analysis of the database LinkProt that collects information about protein chains and complexes that form links. Also, our table can be used to construct table of prime links in lens spaces.

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Classification of Links of Small Complexity in the Thickened Torus

Cited by 8 publications

References 12 publications

Tabulation of Prime Projections of Links in the Thickened Surface of Genus 2 with no more than 4 Crossings

Tabulation of Prime Projections of Links in the Thickened Surface of Genus 2 with no more than 4 Crossings

Classification of links in the thickened Klein bottle

Mathematical modelling of biology processes based on the table of prime links in the solid torus up to 4 crossings

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