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Teoreticheskaya i Matematicheskaya Fizika, 2023, Volume 216, Number 1, Pages 20–35
DOI: https://doi.org/10.4213/tmf10491 (Mi tmf10491) 		 
On an alternative stratification of knots

E. N. Laninaab, A. V. Popolitovabc, N. S. Tselousovab

a Moscow Institute of Physics and Technology (National Research University), Dolgoprudny, Moscow Region, Russia,
b National Research Centre "Kurchatov Institute", Moscow, Russia
c Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow, Russia
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DOI: https://doi.org/10.4213/tmf10491
Abstract: We introduce anЁalternative stratification of knots: by the size of the lattice on which a knot can be first met. Using this classification, we find the fraction of unknots and knots with more than $10$ minimal crossings inside different lattices and answer the question of which knots can be realized inside $3\times 3$ and $5\times 5$ lattices. In accordance with previous research, the fraction of unknots decreases exponentially with the growth of the lattice size. Our computational results are consistent with theoretical estimates for the number of knots with a fixed crossing number inside lattices of a given size.
Keywords: knot theory, knots classification, Jones polynomial, lattice knot.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation 075-15-2019-1619
Foundation for the Development of Theoretical Physics and Mathematics BASIS
Russian Foundation for Basic Research 20-01-00644
21-51-46010-CT_a
21-52-52004_MHT
This work was funded by a grant of the Leonard Euler International Mathematical Institute in Saint Petersburg No. 075-15-2019-1619 (E. L., N. T.), by grants of the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” (E. L., N. T.), by the RFBR grant 20-01-00644 (N. T., A. P.), by the joint RFBR and TUBITAK grant 21-51-46010-CT_a (N. T.) and by the joint RFBR and MOST grant 21-52-52004_MHT (A. P.).
Received: 04.03.2023
Revised: 04.03.2023
English version:
Theoretical and Mathematical Physics, 2023, Volume 216, Issue 1, Pages 924–937
DOI: https://doi.org/10.1134/S0040577923070024
Bibliographic databases:
Document Type: Article
PACS: 02.10. Kn
MSC: 57K10
Language: Russian
Citation: E. N. Lanina, A. V. Popolitov, N. S. Tselousov, “On an alternative stratification of knots”, TMF, 216:1 (2023), 20–35; Theoret. and Math. Phys., 216:1 (2023), 924–937
Citation in format AMSBIB
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\by E.~N.~Lanina, A.~V.~Popolitov, N.~S.~Tselousov
\paper On an alternative stratification of knots
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\vol 216
\issue 1
\pages 20--35
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\crossref{https://doi.org/10.4213/tmf10491}
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\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2023TMP...216..924L}
\transl
\jour Theoret. and Math. Phys.
\yr 2023
\vol 216
\issue 1
\pages 924--937
\crossref{https://doi.org/10.1134/S0040577923070024}
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