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Super-representations of 3-manifolds and torsion polynomials

Stavros Garoufalidis, Seokbeom Yoon

Abstract

Torsion polynomials connect the genus of a hyperbolic knot (a topological invariant) with the discrete faithful representation (a geometric invariant). Using a new combinatorial structure of an ideal triangulation of a 3-manifold that involves edges as well as faces, we associate a polynomial to a cusped hyperbolic manifold that conjecturally agrees with the $\BC^2$-torsion polynomial, which conjecturally detects the genus of the knot. The new combinatorics is motivated by super-geometry in dimension 3, and more precisely by super-Ptolemy assignments of ideally triangulated 3-manifolds and their $\mathrm{OSp}_{2|1}(\BC)$-representations. Extended section 4, and added superalgebras with a single odd generator.

Super-representations of 3-manifolds and torsion polynomials

Abstract

Torsion polynomials connect the genus of a hyperbolic knot (a topological invariant) with the discrete faithful representation (a geometric invariant). Using a new combinatorial structure of an ideal triangulation of a 3-manifold that involves edges as well as faces, we associate a polynomial to a cusped hyperbolic manifold that conjecturally agrees with the -torsion polynomial, which conjecturally detects the genus of the knot. The new combinatorics is motivated by super-geometry in dimension 3, and more precisely by super-Ptolemy assignments of ideally triangulated 3-manifolds and their -representations. Extended section 4, and added superalgebras with a single odd generator.
Paper Structure (21 sections, 10 theorems, 96 equations, 8 figures)

This paper contains 21 sections, 10 theorems, 96 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Torsion polynomials: a Thurstonian connection
  4. Super-geometry in dimension 3
  5. $\mathrm{OSp}_{2|1}(\mathbbm C)$-representations of 3-manifolds
  6. The orthosymplectic group
  7. The unipotent subgroup and pairings
  8. Super-Ptolemy assignments
  9. Natural cocycles
  10. Decorations
  11. Action on $P_{2|1}(\mathcal{T})$
  12. $(m,l)$-deformation
  13. Concrete triangulations
  14. Example: the $4_1$ knot
  15. 1-loop and $\mathbbm C^2$-torsion polynomials
  16. ...and 6 more sections

Key Result

Lemma 2.3

If any two of eqn.odd together with eqn.ptolemy are satisfied, then so are the other two.

Figures (8)

  • Figure 1: From decorated representations to Ptolemy assignments, with the bilinear and trilinear forms as in \ref{['bracket2']} and \ref{['bracket3']}.
  • Figure 2: From Ptolemy assignments to natural cocycles.
  • Figure 3: Edge and face labels for a tetrahedron.
  • Figure 4: Truncating an ideal tetrahedron.
  • Figure 5: A hexagonal face.
  • ...and 3 more figures

Theorems & Definitions (37)

  • Remark 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 27 more