Triangulation independent Ptolemy varieties

Abstract

The Ptolemy variety for ${{\mathrm{SL}}}(2,{\mathbb {C}})$ 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 ${{\mathrm{SL}}}(2,{\mathbb {C}})$-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 ${{\mathrm{SL}}}(2,{\mathbb {C}})$-representations. We also define variants of the Ptolemy variety for ${{\mathrm{PSL}}}(2,{\mathbb {C}})$-representations, and representations that are not necessarily boundary-unipotent. In particular, we obtain an algorithm to compute all irreducible ${{\mathrm{SL}}}(2,{\mathbb {C}})$-characters as well as the full A-polynomial. All the varieties are topological invariants of M.