Topology Seminar 
Friday, April 16, 2010, 11:15--12:15 p.m. in ILB 370

Title: The matrix-tree theorem in knot theory.

Abstract:
The matrix-tree theorem expresses the determinant of some matrix, the 
principal cofactors of the Laplacian of a graph, as a sum over spanning 
trees of the graph. The are some other versions and generalizations of this 
theorem. I will speak about their applications in the theory of links. 
     Namely, the theorems of Hosokawa, Hartley, and Hoste state that for an 
m-component link L the coefficients c_i(L) of the Conway polynomial of L 
vanish when i<m-1 and the coefficient c_{m-1}(L) depends only on the 
linking numbers l_{ij}(L) between the i-th and j-th components of L. This 
coefficient is equal to the determinant of a certain matrix composed of the 
linking numbers from the matrix-tree theorem. The corresponding graph has m 
vertices and its edges are labeled by the linking numbers.
     For virtual links there are two different types of the linking number 
and two Conway polynomials, ascending and descending. Last summer the group 
of my students, Z.Cheng, T.Dokos, and J.Lindquist, generalized the theorem 
above to virtual links. In this case the determinant representing 
c_{m-1}(L) is related to the oriented version of the matrix-tree theorem.