3-manifold polynomials

We propose a way to derive polynomial invariants of closed, orientable $3$-manifolds from Heegaard diagrams via cellularly embedded graphs. Given a Heegaard diagram of an irreducible $3$-manifold $M$, we associate a Heegaard graph $G\subset S$ on the Heegaard surface and restrict to those arising from minimal-genus splittings with a minimal number of vertices. We prove that, up to the natural equivalence of embedded graphs, only finitely many of such minimal Heegaard graphs occur for a fixed manifold. This finiteness enables the definition of $3$-manifold polynomials by evaluating embedded-graph polynomials on representatives of these classes. For lens spaces we show that the associated Heegaard graphs can be fully classified, and that this classification coincides with the classical one for $L(p,q)$. In this setting the Tutte, Penrose, and Bollob\'as-Riordan polynomials behave as invariants of lens spaces, and computational evidence suggests that they may in fact be complete invariants. For the Poincar\'e homology sphere we find that distinct minimal Heegaard diagrams yield coinciding ribbon-graph polynomials, opening the way to interesting conjectures about their discriminating power and the possibility of completeness for broader families of $3$-manifolds.