Degenerations of ideal hyperbolic triangulations

Abstract

Let M be a cusped 3-manifold, and let T be an ideal triangulation of M. The deformation variety D(T ), a subset of which parameterises (incomplete) hyperbolic structures obtained on M using T , is defined and compactified by adding certain projective classes of transversely measured singular codimension-one foliations of M. This leads to a combinatorial and geometric variant of well-known constructions by Culler, Morgan and Shalen concerning the character variety of a 3-manifold.

3-Manifolds and 3d indices

“…Otherwise, it has been proven that all hyperbolic knot complements have nonunivalent triangulations [58], and the same seems to hold even for non-hyperbolic knots [56]. e.g., [28]):…”

Optimistic Limits of the Colored Jones Polynomials

“…By (19) and (20), the product of all shape parameters assigned to the dots is −1, so (17) is the product of shape parameters assigned to the edges C n1 D n1 = B n1 D n1 and some others identified to these. This fact is still true when some of the horizontal edges or nonhorizontal edges of the octahedra are collapsed because of the same reason explained above for the case of (16).…”

“…We call a solution z of H 1 essential if no shape parameters are in {0, 1, ∞}, which implies no edges of the triangulation are homotopically nontrivial. A well-known fact is that if the hyperbolicity equation has an essential solution, then there is a unique geometric solution z (0) of H 1 (for details, see [16,Section 2.8]). Therefore, to guarantee the existence of the geometric solution, Yokota assumed the existence of an essential solution.…”

The 3D index of an ideal triangulation and angle structures

“…A generalized angle structure on T is strict if its restriction to each tetrahedron is strict. For a detailed discussion of angle structures and their duality with normal surfaces, see [11,16,26]. Generalized angle structures are linearizations of the gluing equations, that may be used to construct complete hyperbolic structures, and intimately connected with the theory of normal surfaces on M [12].…”