### Seminars

### on knot theory and related topics

# A method for distingiushing Legendrian and transverse links

Date:

Time:

2021 04 26

15:30 (GMT+0)

Speaker:

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

Join Zoom Meeting

Meeting ID: 818 6674 5751

Passcode: 141592