Tabulating knots in the thickened Klein bottle

Matveev, Sergei; Nabeeva, L. R.

doi:10.1134/s0037446616030174

Cited by 9 publications

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“…Ïîëó÷åííàÿ òàáëèöà ñîäåðaeèò 25 ïîïàðíî íåýêâèâàëåíòíûõ çàöåïëåíèé ñ äâóìÿ è òðåìÿ êîìïîíåíòàìè, ìèíèìàëüíûå äèàãðàììû êîòîðûõ èìåþò íå áîëåå òð¼õ ïåðåêð¼ñòêîâ. Çàöåïëåíèÿ ñ îäíîé êîìïîíåíòîé â äàííîé ðàáîòå Nabeeva íå ðàññìàòðèâàþòñÿ, ïîñêîëüêó òàáóëèðîâàíèå óçëîâ áûëî ïðîâåäåíî â ðàáîòå [6]. Àâòîð áëàãîäàðèò Ñ.Â.…”

Section: ââåäåíèåunclassified

“…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

“…Ðàññìîòðèì ñóùåñòâåííûå ïðîåêöèè, ÿâëÿþùèåñÿ âëîaeåíèåì ýòèõ 6 ãðàôîâ.Äëÿ ïîñòðîåíèÿ îñòàëüíûõ ïðîåêöèé íàì ïîíàäîáèòñÿ äâå îïåðàöèè. Ïåðâàÿ îïåðàöèÿ îïåðàöèÿ óñòðàíåíèÿ ïåðåêð¼ñòêà, ââåä¼ííàÿ â ðàáîòå[6], èçîáðà-ae¼ííàÿ íà ðèñóíêå 5 ñëåâà. Îíà çàêëþ÷àåòñÿ â ñëåäóþùåì: îðèåíòèðóåì ðåáðà ïðîåêöèè, èñïîëüçóÿ ëþáîé èç äâóõ å¼ ïîëíûõ îáõîäîâ ïî ïðàâèëó ¾ïðÿìî âïåð¼ä¿.…”

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Classification of links in the thickened Klein bottle

Nabeeva¹

2020

Sib. Elektron. Mat. Izv.

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Section: ïðåäñòàâëåíèÿ çàöåïëåíèé â óòîëù¼ííîé áóòûëêå êëåéíàunclassified

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Classification of links in the thickened Klein bottle

Nabeeva¹

2020

Sib. Elektron. Mat. Izv.

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“…As regards tabulation of global knots, note that knots in the solid torus [4] and the thickened Klein bottle [5], as well as prime knots in the lens spaces [6] are tabulated. In the knot theory, recent classifications consider only the so-called prime objects, which can not be obtained by some known operations from already tabulated objects.…”

Section: Introductionmentioning

confidence: 99%

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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An Analog of the Kauffman Bracket Polynomial for Knots in the Non-orientable Thickening of a Non-orientable Surface

Tarkaev

2024

Results Math

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Tabulating knots in the thickened Klein bottle

Cited by 9 publications

References 2 publications

Classification of links in the thickened Klein bottle

Classification of links in the thickened Klein bottle

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

An Analog of the Kauffman Bracket Polynomial for Knots in the Non-orientable Thickening of a Non-orientable Surface

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