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- Sergei Chmutov
- J. Comb. Theory, Ser. B
- 2009

We generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed Bollobás-Riordan polynomials of dual graphs. This relation unifies various recent results expressing the Jones polynomial of links as… (More)

- Sergei Chmutov
- 2006

We show that the Kauffman bracket [L] of a checkerboard colorable virtual link L is an evaluation of the Bollobás-Riordan polynomial RGL of a ribbon graph associated with L. This result generalizes the celebrated relation between the classical Kauffman bracket and the Tutte polynomial of planar graphs.

- Sergei Chmutov
- 2004

For a ribbon graph G we consider an alternating link LG in the 3-manifold G× I represented as the product of the oriented surface G and the unit interval I . We show that the Kauffman bracket [LG] is an evaluation of the recently introduced Bollobás-Riordan polynomial RG. This results generalizes the celebrated relation between Kauffman bracket and Tutte… (More)

- Sergei Chmutov, S . V . Duzhin, A . I . KaishevAbstractWe
- 1998

We introduce and study the structure of an algebra in the linear space spanned by all regular 3-valent graphs with a prescribed order of edges at every vertex, modulo certain relations. The role of this object in various areas of low dimensional topology is discussed. 0 Introduction Regular graphs of degree 3, i. e. graphs in which every vertex is incident… (More)

- Sergei Chmutov
- 2007

The Melvin{Morton conjecture says how the Alexander{Conway knot invariant function can be read from the coloured Jones function. It has been proved by D. Bar-Natan and S. Garoufalidis. They reduced the conjecture to a statement about weight systems. The proof of the latter is the most diicult part of their paper. We give a new proof of the statement based… (More)

- Sergei Chmutov, S. DUZHIN
- 1997

This is an overview article on the Kontsevich integral written for the Encyclopedia of Mathematical Physics to be published by Elsevier.

- Sergei Chmutov, S . DuzhinThe
- 1995

We review the explicit formulas for Arnold's generic curve invari-ants due to Viro, Shumakovich and Polyak and add some remarks concerning the invariants of spherical curves and curves immersed into arbitrary orientable surfaces. A generic curve is a smooth immersion of the circle into the plane whose only singularities are transversal double points. Up to… (More)

- Sergei Chmutov
- 2008

- Sergei Chmutov
- 2002

One of the useful methods of the singularity theory, the method of real morsifications [AC0, GZ] (see also [AGV]) reduces the study of discrete topological invariants of a critical point of a holomorphic function in two variables to the study of some real plane curves immersed into a disk with only simple double points of self-intersection. For the closed… (More)

- Sergei Chmutov
- 2003

There are several fascinating relations of plane immersed curves and links. One of them which goes through Legendrian links led Arnold [5] to discovery of three simple invariants J, J−, St of such a curve. N. A’Campo in [2] suggested another construction of a link from a generic immersion of a curve into a 2-disk. It is tightly related to the singularity… (More)