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M.Goussarov (1958--1999), together with of V.Vassiliev and independent
of the latter, was one of the inventors of the notion of finite type invariants
in low-dimensional topology.

In 1994, M.Polyak and O.Viro discovered a way to write out formulas for
finite-type invariants fo r knots and links as closed expressions in terms of
Gauss diagrams of the plane projections. Goussarov proved that any such
invariant can be written in this way.

It turns out that (1) not every linear combination of Gauss diagrams give
rise to an invariant (2) some different combinations lead to one and the same
invariant. The relation between linear combinations and invariants is rather
enigmatic. It has been fully understood only in degree 2. In degree 3, three
different formulas for the basic invariant are known, but nobody understands
the relation between them. In degree 4, several attemps to find the formulas
for the basic invariants were made, but all of them later proved to be false.

I will give an outline of the proof of Goussarov's theorem, speak about the
mentioned problem and quote some recent results of Chmutov with coauthors
who give Gauss-diagram formulas for some serial invariants, viz. coefficients of
the Conway and HOMFLY polynomials.

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http://www.sci.waseda.ac.jp/campus
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For access, see here.

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