arXiv subjects
Publications and source records attributed to Sergei Chmutov.
We show that the Kauffman bracket $[L]$ of a checkerboard colorable virtual link $L$ is an evaluation of the Bollobás-Riordan polynomial $R_{G_L}$ of a ribbon graph associated with $L$. This result generalizes Thistlethwaite's celebrated theorem relating the Kauffman bracket with the Tutte polynomial of planar graphs.
This is an overview article on the Kontsevich integral written for the Encyclopedia of Mathematical Physics, to be published by Elsevier.
For a ribbon graph $G$ we consider an alternating link $L_G$ in the 3-manifold $G\times I$ represented as the product of the oriented surface $G$ and the unit interval $I$. We show that the Kauffman bracket $[L_G]$ is an evaluation of the recently introduced Bollobas-Riordan polynomial $R_G$. This results generalizes the celebrated relation between Kauffman bracket and Tutte polynomial of planar graphs.
A few years ago N.A'Campo invented a construction of a link from a real curve immersed into a disk. In the case of the curve originating from the real morsification method the link is isotopic to the link of the corresponding singularity. There are some curves which do not occur in the singularity theory. In this article we describe the Casson invariant of A'Campo's knots as a J^{+/-}-type invariant of the immersed curves. Thus we get an integral generalization of the Gusein-Zade--Natanzon theorem which says that the Arf invariant of a singularity is equal to J^{-}/2(mod 2) of the corresponding immersed curve. It turns out that this invariant is a second order invariant of the mixed J^{+}- and J^{-}-types.
Recently N.A'Campo suggested a construction of a link from a generic immersion of a curve into a 2-disk. It is tightly related to the singularity theory. In this paper, we give a simple procedure to draw a diagram of the link from a picture of the curve.