Sergei Chmutov

Generalized duality for graphs on surfaces and the signed Bollobás--Riordan polynomial

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 Bollobas-Riordan polynomials of dual graphs. This relation unifies various recent results expressing the Jones polynomial of links as specializations of the Bollobas-Riordan polynomials.

The Kontsevich Integral

The paper contains a detailed exposition of the construction and properties of the Kontsevich integral invariant, crucial in the study of Vassiliev knot invariants.

AN UPPER BOUND FOR THE NUMBER OF VASSILIEV KNOT INVARIANTS

We prove that the number of independent Vassiliev knot invariants of order n is less than (n − 1)! — thus strengthening the a priori bound (2n − 1)!!

THISTLETHWAITE'S THEOREM FOR VIRTUAL LINKS

The celebrated Thistlethwaite theorem relates the Jones polynomial of a link with the Tutte polynomial of the corresponding planar graph. We give a generalization of this theorem to virtual links. In this case, the graph will be embedded into a (higher genus) surface. For such graphs we use the generalization of the Tutte polynomial discovered by B.Bollobas and O.Riordan.

Mutant knots and intersection graphs

We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram itself. The converse statement is easy and well known. We discuss relationship between our results and certain Lie algebra weight systems.

Regular Legendrian knots and the HOMFLY polynomial of immersed plane curves

Abstract. We show that every unframed knot type in

On the Tutte-Krushkal-Renardy polynomial for cell complexes

ST^*{\bf \mathrm{R}}^2

POLYAK–VIRO FORMULAS FOR COEFFICIENTS OF THE CONWAY POLYNOMIAL

has a representative obtained by the Legendrian lifting of an immersed plane curve. This gives a positive answer to the question asked by V.I.Arnold in [3]. The Legendrian lifting lowers the framed version of the HOMFLY polynomial [20] to generic plane curves. We prove that the induced polynomial invariant can be completely defined in terms of plane curves only. Moreover it is a genuine, not Laurent, polynomial in the framing variable. This provides an estimate on the Bennequin-Tabachnikov number of a Legendrian knot.

A lower bound for the number of Vassiliev knot invariants

Recently V. Krushkal and D. Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A. Duval, C. Klivans, and J. Martin. Moreover, after a slight modification, the Tutte-Krushkal-Renardy polynomial evaluated at the origin gives a weighted count of cellular spanning trees, and therefore its free term can be calculated by the cellular matrix-tree theorem of Duval et al. In the case of cell decompositions of a sphere, this modified polynomial satisfies the same duality identity as the original polynomial. We find that evaluating the Tutte-Krushkal-Renardy along a certain line gives the Bott polynomial. Finally we prove skein relations for the Tutte-Krushkal-Renardy polynomial.

Explicit formulas for Arnold’s generic curve invariants

We describe the Polyak–Viro arrow diagram formulas for the coefficients of the Conway polynomial. As a consequence, we obtain the Conway polynomial as a state sum over some subsets of the crossings of the knot diagram. It turns out to be a simplification of a special case of Jaegers state model for the HOMFLY polynomial.