|
Organizers |
Distinguishing Legendrian knots having trivial topological symmetry group
by
Ivan Dynnikov
Steklov Mathematical Institute of Russian Academy of Sciences
Coauthors: Vladimir Shastin (Moscow State University)
Each rectangular diagram of a knot (aka grid diagram or arc-presentation) naturally defines two Legendrian knots, one with respect to the standard contact structure and the other with respect to its mirror image. If K is a topological type of a knot that has no non-trivial symmetry we show that two rectangular diagrams R1 and R2 representing K are related by a finite sequence of exchange moves if and only if the Legendrian knots defined by R1 are equivalent to the respective Legendrian knots defined by R2.
This allows in certain cases to relatively easily distinguish Legendrian knot types that cannot be distinguished by any known algebraic invariants either because the invariants are equal for the knots or because they are too hard to compute. In particular, we prove the existence of an annulus A tangent to the contact structure along the whole boundary, such that the two boundary components of A are not equivalent as Legendrian knots. We use the concrete example of such an annulus that was suggested previously by I.Dynnikov and M.Prasolov.
The work is based on a recent study of rectangular diagrams of surfaces by I.Dynnikov and M.Prasolov.
Date received: April 10, 2017
Copyright © 2017 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbof-07.