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I am interested in Knot Theory and Finite Degree
Invariants. Here are some directions I am trying to pursue.
The Kontsevich integral is a
remarkable universal object allowing one to compute all Vassiliev
finite degree invariants of knots.
There exist two approaches to studying the Kontsevich integral. In the
first approach one fixes the degree of Vassiliev's invariants and
constructs formulas that give their values for all knots.
In 1998 I gave a combinatorial formula for Vassiliev's invariants of
degree 3 (in a new form) and 4. These formulas allow one to compute the
Kontsevich integral modulo 5-degree terms for any knot.
The problem I am interested in is to construct examples of formulas for
finite type invariants of degree greater than 5 and then generalize the
formulas.
In the second approach one fixes a knot (or a knot type) and computes
the Kontsevich integral completely. A known result is the remarkable
formula by D.Bar-Natan, S.Garoufalidis, L.Rozansky, D.Thurston (BGRT)
for the unknot, the formula involving the Bernouli numbers.
Since 1999 I have been studying the problem of computation of the
Kontsevich integral for torus knots. First I obtained an explicit
expression for the Kontsevich integral for the trefoil modulo 7th order
terms. Then I extended this result to all (2,n)-type torus knots. |
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| Natalia Kopteva
D.Bar-Natan
S.Chmutov
S.Duzhin
O.Viro
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Here is the list of my
publications, most of them can be downloaded from the Los Alamos
archive at
http://xxx.lanl.gov/find/math.
| 1. |
Diagrammatic formulae of
Viro-Polyak type for knot
invariants of finite order,
Russian Mathematical Surveys, 1999, 54(3), 658-659.
http://xxx.lanl.gov/abs/math.AT/9905162 |
| 2. |
On formulas of Lannes and Viro-Polyak type for
finite degree invariants, Mat. Zametki 4 (1999), vol. 66. http://xxx.lanl.gov/abs/math.AT/9905160 |
| 3. |
On formulas of the type of Lannes and Viro-Polyak
for Vassiliev invariants, International conference dedicated to the
90th anniversary of L.S.Pontryagin, Moscow, August 31-September 6 1998.
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| 4. |
The transition matrix of the Gauss equations and
the Kontsevich integral for $(2,n)$-torus knots. - Proc. International
conference on differential and functional differential equations,
Moscow, August 1999. |
| 5. |
Diagram formulas of Viro-Polyak type and the
Kontsevich integral for $(2,n)$-torus knots. - Proc. Rokhlin Memorial
conference, Euler Institute, Saint-Petersburg, August 18-25 1999, pdf file. |
| 6. |
On the universal Vassiliev invariant. -
International conference "Geometry and applications" dedicated to the
70th anniversary of V.A.Toponogov, Novosibirsk, March 13-16, 2000. |
| 7. |
(joint with A.Varchenko) A remark on the $sl_2$
approximation of the Kontsevich integral of the unknot. - Preprint, http://xxx.lanl.gov/abs/math.AT/0111201;
- "Zapiski Nauchnyh Seminarov POMI", Vol.299, pp.30-37. ps.gz file
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| 8. |
(joint with A.Varchenko)Finite order
invariants for $(n,2)$-torus knots and
the curve $Y^2=X^3+X^2$. - Preprint,
http://xxx.lanl.gov/abs/math.GT/0402185 |
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| 1. |
International conference dedicated to the 90th
anniversary of L.S.Pontryagin. Moscow September 1998. Talk: On formulas
of the type of Lannes and Viro-Polyak for Vassiliev invariants. |
| 2. |
International Conference "Monodromie et
\'equations diff\'erentielles en th\'eorie des singularit\'es et des
repr\'esentations de groupes." Luminy (France) January 1999. Talk:
Explicit formulas for finite degree invariants. |
| 3. |
International Summer Conference at Chelyabinsk
State University "Low-dimensional Topology and Combinatorial Group
Theory", Chelyabinsk (Russia) July 31 - August 6 1999. Talk: The
Kontsevich integral for $(2,n)$-type torus knots. |
| 4. |
International conference on differential and
functional differential equations, Moscow (Russia) August 1999. Talk:
Connection matrix of KZ-equation and the Kontsevich integral for
$(2,n)$-torus knots. |
| 5. |
Rokhlin Memorial conference, Euler Institute,
Saint-Petersburg (Russia) August 18-25 1999. Talk: Diagram formulas of
Viro-Polyak type and the Kontsevich integral for $(2,n)$-torus knots. |
| 6. |
International conference "Geometry and
applications", Novosibirsk (Russia), March 2000. Talk: Explicit formula
for Kontsevich integral of an arbitrary knot modulo 4-degree terms. |
| 7. |
International Conference "Monodromie et
\'equations diff\'erentielles en th\'eorie des singularit\'es et des
repr\'esentations de groupes. II" Barcelona (Spain) July 2000. Talk:
Drinfeld's associator and the Kontsevich integral. |
| 8. |
The Third European Congress of Mathematics,
Barcelona (Spain) July 2000. Poster: On the Kontsevich integral. |
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