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3-manifold polynomials

José Frías, José Carlos Gómez-Larrañaga, José Luis León-Medina, Fabiola Manjarrez-Gutiérrez

Abstract

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 Bollobás-Riordan polynomials behave as invariants of lens spaces, and computational evidence suggests that they may in fact be complete invariants. For the Poincaré 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.

3-manifold polynomials

Abstract

We propose a way to derive polynomial invariants of closed, orientable -manifolds from Heegaard diagrams via cellularly embedded graphs. Given a Heegaard diagram of an irreducible -manifold , we associate a Heegaard graph 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 -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 . In this setting the Tutte, Penrose, and Bollobás-Riordan polynomials behave as invariants of lens spaces, and computational evidence suggests that they may in fact be complete invariants. For the Poincaré 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 -manifolds.

Paper Structure

This paper contains 13 sections, 17 theorems, 15 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Heegaard diagrams
  4. Cellularly embedded graphs
  5. Embedded graph polynomials
  6. Heegaard graphs
  7. 3-manifold polynomials
  8. The case of lens spaces
  9. Lens spaces polynomials
  10. The Tutte and Bollobás--Riordan polynomials
  11. The Penrose polynomial
  12. A first look at the Poincaré sphere
  13. Conclusion

Key Result

Theorem 7

Let $G$ be an embedded graph and take $k\in \mathbb{N}$, then $P(G;k)=\sum (-1)^{cr(s)}$, where the sum is taken over all admissible $k$-valuations $s$ of $G_m$ and $cr(s)$ is the number of crossing states in $s$.

Figures (10)

  • Figure 1: A genus–two Heegaard diagram for the Poincaré sphere $P_3$ with 12 vertices
  • Figure 2: Construction of the medial graph
  • Figure 3: Admissible $k$-valuation
  • Figure 4: Homeomorphic lens spaces
  • Figure 5: Homeomorphic lens spaces
  • ...and 5 more figures

Theorems & Definitions (48)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 7: ELLISMONAGHAN2013424
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 38 more