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Distinguishing Legendrian knots with trivial orientation-preserving symmetry group

Ivan Dynnikov and Vladimir Shastin

Algebraic & Geometric Topology 23 (2023) 1849–1889

Recent work of I Dynnikov and M Prasolov proposes a new method of comparing Legendrian knots. In general, to apply the method requires a lot of technical work. In particular, one needs to search all rectangular diagrams of surfaces realizing certain dividing configurations. We show that in the case when the orientation-preserving symmetry group of the knot is trivial, this exhaustive search is not needed, which simplifies the procedure considerably. This allows one to distinguish Legendrian knots in certain cases when the computation of the known algebraic invariants is infeasible or not informative. In particular, we disprove that when A 3 is an annulus tangent to the standard contact structure along A, then the two components of A are always equivalent Legendrian knots. A candidate counterexample was proposed recently by Dynnikov and Prasolov, but the proof of the fact that the two components of A are not Legendrian equivalent was not given. Now this work is accomplished. It is also shown here that the problem of comparing two Legendrian knots having the same topological type is algorithmically solvable provided that the orientation-preserving symmetry group of these knots is trivial.

Legendrian knot, rectangular diagram
Mathematical Subject Classification
Primary: 57K10, 57K33
Received: 27 March 2021
Revised: 30 July 2021
Accepted: 10 September 2021
Published: 14 June 2023
Ivan Dynnikov
V A Steklov Mathematical Institute
Russian Academy of Science
St Petersburg State University
Saint Petersburg
Vladimir Shastin
Department of Mechanics and Mathematics
Moscow State University
St Petersburg State University
Saint Petersburg

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