A calculus for ideal triangulations of three‐manifolds with embedded arcs

Refining the notion of an ideal triangulation of a compact three‐manifold, we provide in this paper a combinatorial presentation of the set of pairs (M,α), where M is a three‐manifold and α is a collection of properly embedded arcs. We also show that certain well‐understood combinatorial moves are sufficient to relate to each other any two refined triangulations representing the same (M,α). Our proof does not assume the Matveev–Piergallini calculus for ideal triangulations, and actually easily implies this calculus. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

[1]  J. W. Alexander,et al.  The Combinatorial Theory of Complexes , 1930 .

[2]  B. G. Casler.  An imbedding theorem for connected 3-manifolds with boundary , 1965 .

[3]  Sergei Matveev,et al.  Constant energy surfaces of Hamiltonian systems, enumeration of three-dimensional manifolds in increasing order of complexity, and computation of volumes of closed hyperbolic manifolds , 1988 .

[4]  Riccardo Piergallini,et al.  Standard moves for standard polyhedra and spines , 1988 .

[5]  Sergei Matveev,et al.  TRANSFORMATIONS OF SPECIAL SPINES AND THE ZEEMAN CONJECTURE , 1988 .

[6]  Udo Pachner,et al.  P.L. Homeomorphic Manifolds are Equivalent by Elementary 5hellingst , 1991, Eur. J. Comb..

[7]  Vladimir Turaev,et al.  State sum invariants of 3 manifolds and quantum 6j symbols , 1992 .

[8]  A. Makovetskii,et al.  On transformations of special spines and special polyhedra , 1999 .

[9]  C. Petronio,et al.  Construction and recognition of hyperbolic 3-manifolds with geodesic boundary , 2001, math/0109012.

[10]  QHI, 3-manifolds scissors congruence classes and the volume conjecture , 2002, math/0211053.

[11]  R. Benedetti,et al.  Quantum hyperbolic invariants of 3-manifolds with PSL(2; C)-characters , 2004 .

[12]  QHI Theory, I: 3-Manifolds Scissors Congruence Classes and Quantum Hyperbolic Invariants , 2002, math/0201240.

[13]  Sergei Matveev,et al.  Algorithmic Topology and Classification of 3-Manifolds , 2003 .

[14]  Classical and quantum dilogarithmic invariants of flat PSL(2,C) -bundles over 3-manifolds , 2003, math/0306283.