Super-representations of 3-manifolds and torsion polynomials

TL;DR

This paper introduces a novel combinatorial approach to associate polynomials with cusped hyperbolic 3-manifolds, linking topological and geometric invariants and motivated by super-geometry and super-Ptolemy assignments.

Contribution

It develops a new combinatorial structure involving edges and faces of ideal triangulations, conjecturally matching torsion polynomials and detecting knot genus.

Findings

Proposes a polynomial associated with hyperbolic 3-manifolds.

Conjecturally matches the $c^2$-torsion polynomial.

Connects super-geometry with 3-manifold invariants.

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.