In this paper we define a new invariant of the incomplete hyperbolic structures on a 1–cusped finite volume hyperbolic 3–manifold , called the ortholength invariant. We show that away from a (possibly empty) subvariety of excluded values this invariant both locally parameterises equivalence classes of hyperbolic structures and is a complete invariant of the Dehn fillings of which admit a hyperbolic structure. We also give an explicit formula for the ortholength invariant in terms of the traces of the holonomies of certain loops in . Conjecturally this new invariant is intimately related to the boundary of the hyperbolic Dehn surgery space of .
"A new invariant on hyperbolic Dehn surgery space." Algebr. Geom. Topol. 2 (1) 465 - 497, 2002. https://doi.org/10.2140/agt.2002.2.465