Exchange classes of rectangular diagrams, Legendrian knots, and the knot symmetry group

Maxim Prasolov

Knots in Washington XLVI: 70th Birthday of Oleg Viro;
May 4-6, 2018
George Washington University,
Washington, DC, USA

Organizers
Valentina Harizanov (GWU), Jozef H. Przytycki (GWU), Yongwu Rong (GWU), Radmila Sazdanovic (NCSU), Alexander Shumakovitch (GWU)

Conference Homepage

Exchange classes of rectangular diagrams, Legendrian knots, and the knot symmetry group
by
Maxim Prasolov
Moscow State University
Coauthors: Ivan Dynnikov (Steklov Mathematical Institute)

Rectangular diagrams are a particularly nice way to represent knots and links in the three-space. The crucial property of this presentation is the existence of a monotonic simplification algorithm for recognizing the unknot [I.D., 2006]. The present research is motivated by an attempt to extend the monotonic simplification procedure to arbitrary knot types.

Another nice feature of rectangular diagrams is their relation to Legendrian knots. Namely, each rectangular diagram defines, in a very natural way, two Legendrian knots, one with respect to the standard contact structure, and the other with respect to the mirror image of the standard contact structure. These two Legendrian knots always have an important mutual independence property [I.D., M.Prasolov, 2013], which is roughly this: any Legendrian stabilization and destabilization of each of the two Legendrian types can be done without altering the other, by applying elementary moves to the rectangular diagram.

Among elementary moves defined for rectangular diagrams, there are those that preserve both Legendrian knot types associated with the diagram. These are exchange moves. An exchange class is a set of rectangular diagrams that can be obtained from a fixed diagram by exchange moves.

Let K be a topological knot type, and let L₁ (respectively, L₂) be a ξ₊-Legendrian (respectively, ξ_--Legendrian) knot type of topological type K, where ξ₊ and ξ_- are the standard contact structure and its mirror image, respectively. There are symmetry groups G, H₁, H₂ naturally associated with K, L₁, and L₂, respectively. We show that the set of exchange classes representing L₁ and L₂ simultaneously, is in one-to-one correspondence with the set H₁\G/H₂ of double cosests.

The proof uses, among other things, a trick from a joint work of I.Dynnikov and V.Shastin (in preparation).

Date received: April 22, 2018