Abstract
The notion of Gem–Matveev complexity (GM-complexity) has been introduced within crystallization theory, as a combinatorial method to estimate Matveev's complexity of closed 3-manifolds; it yielded upper bounds for interesting classes of such manifolds. In this paper, we extend the definition to the case of non-empty boundary and prove that for each compact irreducible and boundary-irreducible 3-manifold it coincides with the modified Heegaard complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via GM-complexity, we obtain an estimation of Matveev's complexity for all Seifert 3-manifolds with base 𝔻2and two exceptional fibers and, therefore, for all torus knot complements.
Highlights
The Matveev’s complexity is a well-known invariant for 3-manifolds, defined in [28] as the minimum number of true vertices among all almost simple spines of the manifold
The idea of estimating c(M ) by using Heegaard decompositions is already suggested in the foundational paper [28]: if H = (S, v, w) is a Heegaard diagram of M, an upper bound for c(M ) is provided by the almost simple spines of M obtained by adding to the surface S the meridian disks corresponding to the systems of curves v and w and by removing the 2-disk corresponding to an arbitrary region Rof S \ (v ∪ w)
The aim of the present paper is to extend the definition of Gem-Matveev complexity to the case of 3-manifolds with non-empty boundary (Section 4), and to prove that GM -complexity and HM complexity turn out to be useful different tools to compute the same upper bound for Matveev’s complexity, in the whole setting of compact irreducible and boundary irreducible 3-manifolds
Summary
The Matveev’s complexity is a well-known invariant for 3-manifolds, defined in [28] as the minimum number of true vertices among all almost simple spines of the manifold. The 3-sphere, the real projective space, the lens space L(3, 1) and the spherical bundles S1 × S2 and S1× ̃ S2 have complexity zero by definition Apart from these special cases, the Matveev’s complexity c(M ) of a closed prime 3-manifold M turns out to be the minimum number of tetrahedra needed to obtain M via face paring of them (see [28, Proposition 2], together with the related Remark). The aim of the present paper is to extend the definition of Gem-Matveev complexity to the case of 3-manifolds with non-empty boundary (Section 4), and to prove that GM -complexity and HM complexity turn out to be useful different tools to compute the same upper bound for Matveev’s complexity, in the whole setting of compact irreducible and boundary irreducible 3-manifolds. In some particular cases (including for example, three torus knot complements with complexity one and the orientable I-bundle over the Klein bottle, having complexity zero), the obtained estimation turns out to coincide with the exact value of Matveev’s complexity (see Corollary 14)
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