Is there a simple formula to compute the Casson invariant of an homology $3$-sphere from its Heegaard diagram?

Question

Let $(S_g,\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2)$ be a Heegaard diagram of a Heegaard splitting $\Sigma=H_g \cup_{\phi_1\phi_2^{-1}}H_g$ of an integral homology sphere $\Sigma$, i.e. $S_g$ is a closed, orientable surface of genus $g$, $H_g$ is the $3$-dimensional $1$-handlebody of genus $g$, $\phi_1,\phi_2 \colon \partial H_g \rightarrow S_g$ are homeomorphisms and $\boldsymbol{\alpha}_i=\phi_i(\boldsymbol{\alpha})$ for $i=1,2$, where $\boldsymbol{\alpha}$ is the set of boundaries of a maximal system of meridians for $H_g$. I know that the Casson invariant of $\Sigma$ can be defined in terms of its Heegaard splitting, but I do not see clearly how to compute it from a Heegaard diagram. Is there a simple way to do that?

Answer 1

I don’t know a nice recipe from a Heegaard diagram, but every Heegaard splitting of a homology 3-sphere is equivalent to one whose gluing map lies in the Torelli subgroup of the mapping class group (ie the subgroup acting trivially on homology), and Morita has a beautiful formula for the Casson invariant in terms of this gluing map in his paper

S. Morita, On the structure of the Torelli group and the Casson invariant, Topology 30 (1991), no. 4, 603–621.

Answer 2

I don't know if this qualifies as a simple formula, but here at least is a procedure. You can convert the Heegaard diagram into a framed link description of your homology sphere, and then use Hoste's formula (A formula for Casson's invariant. Trans. Amer. Math. Soc. 297 (1986), no. 2, 547–562). There is an intermediate step, because Hoste requires that your surgery description have all linking numbers 0, so you have to arrange that first.