From Spanning Trees to Quasitrees
by
Neal Stoltzfus
Louisiana State University

Quasi-trees are connected spanning subgraphs of a graph embedded on a surface

for which the boundary of a regular neighborhood has a single component (a circle.)

In the case of a plane graph, the nghd. is a disc and the subgraph is a tree. We will

describe a method to construct graphs embedded on surfaces from a link

diagrams.

In joint work with Dasbach, Futer, Kalfagianni and Lin, we can compute the Jones

polynomial of a link from the Bollobas-Riordan generalization of the Tutte

polynomial as well as a geometric interpretation of the determinant of a

non-alternating knot. In joint work with Kofman and Champanerkar , we can

generalize Tutte's spanning tree expansion of his polynomial to a quasi-tree

expansion of the polynomial of Bollobas-Riordan and give a quasi-tree basis for

Khovanov homology.

Date received: May 29, 2007