Lectures of Prof Duzhin

## Duzhin 氏の連続講義

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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.