Lectures of Prof Duzhin

## Duzhin 氏の連続講義

**講演者の Duzhin 氏が怪我で通院中のため、連続講義の日程を、**

先にアナウンスしました日程から下記の日程に変更いたしました。

よろしくお願いいたします。

日時：9月28日(木)、29日(金)
各日とも 14:00〜17:00

場所：京都大学 数理解析研究所
402号室(9月28日)、115号室(9月29日)

講演者: Sergei Duzhin 氏 (Steklov Institute)

Title: Detecting the link orientation

Abstract:

I will speak about the problem of detecting the orientation
of knots and links, i.e. finding the invariants that take distinct values
on two links differing only by an inversion.

The first result in this direction is a classical theorem of Trotter
who proved that the pretzel knot P_{3,5,7} is not equivalent to its inverse.
Trotter and some subsequent authors used homomorphisms of the knot group
to study the invertibility.
It is known that knot polynomials obtained by the Reshetikhin--Turaev
procedure do not feel the orientation.
Finite type (Vassiliev) knot invariants are strictly stronger than quantum
invariants, and there is an important problem if these can tell a knot from
its inverse.
This problem is open until now.

For links with more than one component the corresponding problem
is partially solved, namely, a positive answer was obtained for closed links
with 6 or more components (X.-S.Lin) and for string links with 2 components
(S.Duzhin--M.Karev).

I will give a review of the known results on the problem and then speak
about an attempt to solve it for closed 2-component links using
the invariants with values in the necklace algebra.

Another approach that might lead to orientation-detecting invariants is
the categorification of the gl_n Lie algebra weight system. Indeed, the
weight system with values in the center of the universal enveloping algebra
for gl_n is a function on the space of Jacobi diagrams which is given by the
alternating sum over a cube of resolutions of the triple points of a
diagram. One may therefore try to apply the general scheme of
categorification due to Khovanov (through Frobenius algebras) and Bar-Natan
(through canopolies) -- in a hope to split the big alternating sum
into several smaller sums each of which gives a weight system,
whose totality is a stronger invariant than the original weight system.
This is an unfinished project, and I will speak about it in a hope to find
eventual collaborators among the listeners.

連絡先：大槻 知忠
（京大数理研、tomotadakurims.kyoto-u.ac.jp）