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On an alternative stratification of knots

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Abstract

We introduce an alternative 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.

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Acknowledgments

We thank A. Malyutin and Yu. Belousov for the useful discussions.

Funding

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.).

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Authors and Affiliations

  1. Moscow Institute of Physics and Technology (National Research University), Dolgoprudny, Moscow Region, Russia

    E. N. Lanina, A. V. Popolitov & N. S. Tselousov

  2. National Research Center “Kurchatov Institute”, Moscow, Russia

    E. N. Lanina, A. V. Popolitov & N. S. Tselousov

  3. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia

    A. V. Popolitov

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  1. E. N. Lanina
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  3. N. S. Tselousov
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Correspondence to E. N. Lanina.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 216, pp. 20–35 https://doi.org/10.4213/tmf10491.

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Lanina, E.N., Popolitov, A.V. & Tselousov, N.S. On an alternative stratification of knots. Theor Math Phys 216, 924–937 (2023). https://doi.org/10.1134/S0040577923070024

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