  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.
