Complexity computation for compact 3-manifolds via crystallizations and Heegaard diagrams
arXiv:1203.0183 · doi:10.1016/j.topol.2012.05.016
Abstract
The idea of computing Matveev complexity by using Heegaard decompositions has been recently developed by two different approaches: the first one for closed 3-manifolds via crystallization theory, yielding the notion of Gem-Matveev complexity; the other one for compact orientable 3-manifolds via generalized Heegaard diagrams, yielding the notion of modified Heegaard complexity. In this paper we extend to the non-orientable case the definition of modified Heegaard complexity and prove that for closed 3-manifolds Gem-Matveev complexity and modified Heegaard complexity coincide. Hence, they turn out to be useful different tools to compute the same upper bound for Matveev complexity.
12 pages; accepted for publication in Topology and Its Applications, volume containing Proceedings of Prague Toposym 2011