A method for distingiushing Legendrian and transverse links



2021 04 26

15:30 (GMT+0)


Ivan Dynnikov

Knots and representation theory

Legendrian (respectively, transverse) links are smooth links in the
three-space that are tangent (respectively, transverse) to the standard
contact structure. Deciding whether two such links are equivalent modulo a
contactomorphism is a hard problem in general. Many topological invariants
of Legendrian and transverse links are known, but they do not suffice for
a classification even in the case of knots of crossing number six.

In recent joint works with Maxim Prasolov and Vladimir Shastin we
developed a rectangular diagram machinery for surfaces and links in the
three-space. This machinery has a tight connection with contact topology,
namely with Legendrian links and Giroux's convex surfaces. We are mainly
interested in studying rectangular diagrams of links that cannot be
monotonically simplified by means of elementary moves. It turns out that
this question is nearly equivalent to classification of Legendrian links.

The main outcome we have so far is an algorithm for comparing two
Legendrian (or transverse) links. The computational complexity of the
algorithm is, of course, very high, but, in many cases, certain parts of
the procedure can be bypassed, which allows us to distinguish quite
complicated Legendrian knots. In praticular, we have managed to provide an
example of two inequivalent Legendrian knots cobounding an annulus tangent
to the standrard contact structure along the entire boundary. Such
examples were previously unknown.

The work is supported by the Russian Science Foundation under
grant 19-11-00151

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