M.Sc Thesis
M.Sc Student Epstein Baruch A Combinatorial Encoding of 3-Manifolds Department of Mathematics Professor Michael Polyak

Abstract

The Casson invariant is a celebrated integer-valued invariant of 3-manifolds, introduced in 1985 by A.Casson, generalized by K.Walker and later by C.Lescop, and studied extensively since then.

Various formulas and descriptions of this invariant exist, but unfortunately they are quite complicated.

In 2014, M. Polyak used a novel combinatorial encoding of 3-manifolds by weighted graphs to propose a simple formula for the Casson-Walker invariant in terms of counting certain (Theta-shaped and figure-eight) subgraphs.

We review the necessary topological and combinatorial preliminaries: knot theory, link surgery, Kirchoff's matrix tree theorem and its generalizations.

Using these tools, we present M. Polyak's construction, prove some of its properties, describe his formula and relate it to the Casson-Walker invariant using work by C. Lescop.

We study the resulting combinatorial invariant and find an explicit formula for a special case, as well as a recursive relation the invariant satisfies.

Using the Laplace operator on graphs, we present an alternative simplified formula for this invariant via counting of cycle-rooted trees.

Finally, we discuss a few directions to continue and possibly generalize our work.