Geodesic
On an alternative stratification of knots
Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 1, pp. 20-35 Cet article a éte moissonné depuis la source Math-Net.Ru

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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, lattice knot.
Mots-clés : Jones polynomial 
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E. N. Lanina; A. V. Popolitov; N. S. Tselousov. On an alternative stratification of knots. Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 1, pp. 20-35. http://geodesic.mathdoc.fr/item/TMF_2023_216_1_a1/

[1] E. Guadagnini, M. Martellini, M. Mintchev, “Chern–Simons holonomies and the appearance of quantum groups”, Phys. Lett. B, 235:3–4 (1990), 275–281 | DOI | MR

[2] N. Yu. Reshetikhin, V. G. Turaev, “Ribbon graphs and their invariants derived from quantum groups”, Commun. Math. Phys., 127:1 (1990), 1–26 | DOI | MR | Zbl

[3] A. Morozov, A. Smirnov, “Chern–Simons teory in the temporal gauge and knot invariants through the universal quantum $R$-matrix”, Nucl. Phys. B, 835:3 (2010), 284–313, arXiv: 1001.2003 | DOI | MR | Zbl

[4] R. K. Kaul, T. R. Govindarajan, “Three-dimensional Chern–Simons theory as a theory of knots and links”, Nucl. Phys. B, 380:1–2 (1992), 293–333, arXiv: hep-th/9111063 | DOI | MR | Zbl

[5] P. Rama Devi, T. R. Govindarajan, R. K. Kaul, “Three-dimensional Chern–Simons theory as a theory of knots and links. (III). Compact semi-simple group”, Nucl. Phys. B, 402:1–2 (1993), 548–566, arXiv: ; “Knot invariants from rational conformal field theories”, Nucl. Phys. B, 422:1–2 (1994), 291–306, arXiv: hep-th/9212110hep-th/9312215 | DOI | MR | Zbl | DOI | MR | Zbl

[6] P. Ramadevi, T. Sarkar, “On link invariants and topological string amplitudes”, Nucl. Phys. B, 600:3 (2001), 487–511, arXiv: hep-th/0009188 | DOI | MR | Zbl

[7] Zodinmawia, P. Ramadevi, “$SU(N)$ quantum Racah coefficients and non-torus links”, Nucl. Phys. B, 870:1 (2013), 205–242, arXiv: ; Reformulated invariants for non-torus knots and links, arXiv: 1107.39181209.1346 | DOI | MR

[8] S. Nawata, P. Ramadevi, Zodinmawia, “Colored Kauffman homology and Super-A-polynomials”, JHEP, 01 (2014), 126, 69 pp., arXiv: 1310.2240 | DOI | MR

[9] J. Gu, H. Jockers, “A note on colored HOMFLY polynomials for hyperbolic knots from WZW models”, Commun. Math. Phys., 338:1 (2015), 393–456, arXiv: 1407.5643 | DOI | MR

[10] D. Sho, Exchange relation in $sl_3$ WZNW model in semiclassical limit, arXiv: 1408.2212

[11] A. Mironov, A. Morozov, And. Morozov, “Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid”, JHEP, 03 (2012), 034, 33 pp., arXiv: 1112.2654 | DOI | MR

[12] H. Itoyama, A. Mironov, A. Morozov, And. Morozov, “Character expansion for HOMFLY polynomials III: All 3-strand braids in the first symmetric representation”, Internat. J. Modern Phys. A, 27:19 (2012), 1250009, 85 pp. ; “Eigenvalue hypothesis for Racah matrices and HOMFLY polynomials for 3-strand knots in any symmetric and antisymmetric representations”, 28:3–4 (2013), 1340009, 81 pp. | DOI | MR | DOI | MR

[13] A. Anokhina, A. Mironov, A. Morozov, And. Morozov, “Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux”, Adv. High Energy Phys., 2013, 931830, 12 pp. | MR | Zbl

[14] A. S. Anokhina, A. A. Morozov, “Protsedura kablirovaniya dlya raskrashennykh polinomov KhOMFLI”, TMF, 178:1 (2014), 3–68, arXiv: 1307.2216 | DOI | DOI | MR | Zbl

[15] P. Ramadevi, T. R. Govindarajan, R. K. Kaul, “Chirality of knots $9_{42}$ and $10_{71}$ and Chern–Simons theory”, Modern Phys. Lett. A, 9:34 (1994), 3205–3217 | DOI | MR

[16] S. Nawata, P. Ramadevi, Zodinmawia, “Colored HOMFLY polynomials from Chern–Simons theory”, J. Knot Theory Ramifications, 22:13 (2013), 1350078, 58 pp. | DOI | MR | Zbl

[17] D. Galakhov, D. Melnikov, A. Mironov, A. Morozov, A. Sleptsov, “Colored knot polynomials for arbitrary pretzel knots and links”, Phys. Lett. B, 743 (2015), 71–74 | DOI | MR | Zbl

[18] D. Galakhov, D. Melnikov, A. Mironov, A. Morozov, A. Sleptsov, “Knot invariants from Virasoro related representation and pretzel knots”, Nucl. Phys. B, 899 (2015), 194–228 | DOI | MR | Zbl

[19] A. Mironov, A. Morozov, A. Sleptsov, “Colored HOMFLY polynomials for the pretzel knots and links”, JHEP, 07 (2015), 069, 34 pp. | DOI | MR

[20] C. Even-Zohar, J. Hass, N. Linial, T. Nowik, “Universal knot diagrams”, J. Knot Theory Ramifications, 28:07 (2019), 1950031, 30 pp. | DOI | MR

[21] “The Knot Atlas”, http://katlas.org

[22] Yu. S. Belousov, A. V. Malyutin, Hyperbolic knots are not generic, arXiv: 1908.06187

[23] C. C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, Freeman, New York, 1994 | MR

[24] C. Ernst, D. W. Sumners, “The growth of the number of prime knots”, Math. Proc. Cambridge Philos. Soc., 102:2 (1987), 303–315 | DOI | MR

[25] V. F. R. Jones, “A polynomial invariant for knots via von Neumann algebras”, Bull. Amer. Math. Soc. (N. S.), 12:1 (1985), 103–111 | DOI | MR | Zbl

[26] V. G. Turaev, “The Yang–Baxter equation and invariants of links”, New Developments in the Theory of Knots, Advanced Series in Mathematical Physics, 11, ed. T. Kohno, 1990, 175–201 | DOI

[27] V. F. R. Jones, “On knot invariants related to some statistical mechanical models”, Pacific J. Math., 137:2 (1989), 311–334 | DOI | MR | Zbl

[28] R. Bekster, Tochno reshaemye modeli v statisticheskoi mekhanike, Mir, M., 1985 | MR | MR | Zbl

[29] H. E. Lieb, F. Y. Wu, “Two-dimensional ferroelectric models”, Phase Transitions and Critical Phenomena, v. 11, eds. C. Domb, M. S. Green, Academic Press, London, 1972, 331–490

[30] A. Grosberg, S. Nechaev, “Algebraic invariants of knots and disordered Potts model”, J. Phys. A, 25:17 (1992), 4659–4672 | DOI | MR

[31] S. Nechaev, “Statistics of knots and entangled random walks”, Aspects topologiques de la physique en basse dimension/ Topological Aspects of Low Dimensional Systems (NATO Advanced Study Institute, Grenoble, France, Les Houches, Session LXIX, 7–31 July, 1998), Les Houches – Ecole d'Ete de Physique Theorique (LHSUMMER), 69, eds. A. Comtet, T. Jolicoeur, S. Ouvry, F. David, Springer, Berlin, 1999, 643–733 | DOI | MR

[32] M. Khovanov, “A categorification of the Jones polynomial”, Duke Math. J., 101:3 (2000), 359–426 | DOI | MR | Zbl

[33] D. Bar-Natan, “On Khovanov's categorification of the Jones polynomial”, Algebr. Geom. Topol., 2:1 (2002), 337–370 | DOI | MR | Zbl

[34] “KnotInfo: Table of Knots”, https://knotinfo.math.indiana.edu

[35] P. B. Kronheimer, T. S. Mrowka, “Khovanov homology is an unknot-detector”, Publ. Math. IHES, 113:1 (2011), 97–208 | DOI | MR