Seminar on low-dimensional mathematics
June 17, 2005
S.Duzhin
Detecting link orientation by finite type invariants.
This talk is based on a joint work with my student M.Karev.
It is well known that classical knot polynomials and quantum invariants in
general take equal values on a knot and its inverse.
The class of Vassiliev invariants is strictly wider, and the problem whether
they can be used to tell a knot from its inverse is open and seemingly
difficult.
In this talk, we discuss the corresponding problem for links with more than
1 component.
We prove the existence of a degree 7 Vassiliev invariant of long (or string)
two-component links which is not preserved under the simultaneous change of
orientation of both components. The non-invertibility of this invariant
can be detected by the standard weight system with values in the tensor
square of the universal enveloping algebra for $\gl(n)$.
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