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 Uspekhi Mat. Nauk, 2003, Volume 58, Issue 6(354), Pages 45–92 (Mi umn675)

Recognition algorithms in knot theory

I. A. Dynnikov

M. V. Lomonosov Moscow State University

Abstract: In this paper the problem of constructing algorithms for comparing knots and links is discussed. A survey of existing approaches and basic results in this area is given. In particular, diverse combinatorial methods for representing links are discussed, the Haken algorithm for recognizing a trivial knot (the unknot) and a scheme for constructing a general algorithm (using Haken's ideas) for comparing links are presented, an approach based on representing links by closed braids is described, the known algorithms for solving the word problem and the conjugacy problem for braid groups are described, and the complexity of the algorithms under consideration is discussed. A new method of combinatorial description of knots is given together with a new algorithm (based on this description) for recognizing the unknot by using a procedure for monotone simplification. In the conclusion of the paper several problems are formulated whose solution could help to advance towards the “algorithmization” of knot theory.

DOI: https://doi.org/10.4213/rm675

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English version:
Russian Mathematical Surveys, 2003, 58:6, 1093–1139

Bibliographic databases:

UDC: 515.162.8
MSC: Primary 57M25; Secondary 20F10, 20F36, 20F05, 68Q25

Citation: I. A. Dynnikov, “Recognition algorithms in knot theory”, Uspekhi Mat. Nauk, 58:6(354) (2003), 45–92; Russian Math. Surveys, 58:6 (2003), 1093–1139

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/umn675
• https://doi.org/10.4213/rm675
• http://mi.mathnet.ru/eng/umn/v58/i6/p45

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This publication is cited in the following articles:
1. Chernavsky A.V., Leksine V.P., “Unrecognizability of manifolds”, Ann. Pure Appl. Logic, 141:3 (2006), 325–335
2. Funar L., Kapoudjian Ch., “The braided Ptolemy-Thompson group is finitely presented”, Geom. Topol., 12:1 (2008), 475–530
3. Hayashi Ch., Yamada S., “Unknotting Rectangular Diagrams of the Trivial Knot by Exchange Moves”, J. Knot Theory Ramifications, 22:11 (2013), 1350067
4. Maxim Prasolov, “Rectangular diagrams of Legendrian graphs”, J. Knot Theory Ramifications, 23:13 (2014), 1450074
5. Ando T., Hayashi Ch., Hayashi M., “Rectangular Seifert Circles and Arcs System”, J. Knot Theory Ramifications, 23:8 (2014), 1450041
6. Ando T., Hayashi Ch., Nishikawa Yu., “Realizing Exterior Cromwell Moves on Rectangular Diagrams By Reidemeister Moves”, J. Knot Theory Ramifications, 23:5 (2014), 1450023
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