Triangulation Independent Ptolemy Varieties

TL;DR

This paper introduces a triangulation-independent Ptolemy variety that captures all boundary-unipotent irreducible SL(2,C)-representations of a 3-manifold, providing a comprehensive invariant and computational tools.

Contribution

It defines a new Ptolemy variety invariant of 3-manifolds that detects all relevant representations, overcoming triangulation dependence and enabling computation of characters and A-polynomials.

Findings

Defines a triangulation-independent Ptolemy variety.

Provides an algorithm to compute all irreducible SL(2,C)-characters.

Enables calculation of the full A-polynomial.

Abstract

The Ptolemy variety for SL(2,C) is an invariant of a topological ideal triangulation of a compact 3-manifold M. It is closely related to Thurston's gluing equation variety. The Ptolemy variety maps naturally to the set of conjugacy classes of boundary-unipotent SL(2,C)-representations, but (like the gluing equation variety) it depends on the triangulation, and may miss several components of representations. In this paper, we define a Ptolemy variety, which is independent of the choice of triangulation, and detects all boundary-unipotent irreducible SL(2,C)-representations. We also define variants of the Ptolemy variety for PSL(2,C)-representations, and representations that are not necessarily boundary-unipotent. In particular, we obtain an algorithm to compute all irreducible SL(2,C)-characters as well as the full A-polynomial. All the varieties are topological invariants of M.