A note about complexity of lens spaces

Abstract

Within crystallization theory, (Matveev's) complexity of a 3-manifold can be estimated by means of the combinatorial notion of GM-complexity. In this paper, we prove that the GM-complexity of any lens space L(p,q), with p greater than 2, is bounded by S(p,q)-3, where S(p,q) denotes the sum of all partial quotients in the expansion of q/p as a regular continued fraction. The above upper bound had been already established with regard to complexity; its sharpness was conjectured by Matveev himself and has been recently proved for some infinite families of lens spaces by Jaco, Rubinstein and Tillmann. As a consequence, infinite classes of 3-manifolds turn out to exist, where complexity and GM-complexity coincide. Moreover, we present and briefly analyze results arising from crystallization catalogues up to order 32, which prompt us to conjecture, for any lens space L(p,q) with p greater than 2, the following relation: k(L(p,q)) = 5 + 2 c(L(p,q)), where c(M) denotes the complexity of a 3-manifold M and k(M)+1 is half the minimum order of a crystallization of M.

14 pages, 2 figures; v2: we improved the paper (changes in Proposition 10; Corollary 9 and Proposition 11 added) taking into account Theorem 2.6 of arxiv:1310.1991v1 which makes use of our Prop. 6(b) (arxiv:1309.5728v1). Minor changes have been done, too, in particular to make references more essential

References in corpus (3)

• The average dual surface of a cohomology class and minimal simplicial decompositions of infinitely many lens spaces
• Complexity computation for compact 3-manifolds via crystallizations and Heegaard diagrams
• Minimal crystallizations of 3-manifolds

Cited by in corpus (10)

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• 4-colored graphs and knot/link complements
• PL 4-manifolds admitting simple crystallizations: framed links and regular genus
• The average dual surface of a cohomology class and minimal simplicial decompositions of infinitely many lens spaces
• Kirby diagrams and 5-colored graphs representing compact 4-manifolds
• Minimal crystallizations of 3-manifolds
• Cataloguing PL 4-manifolds by gem-complexity
• Minimal 4-colored graphs representing an infinite family of hyperbolic 3-manifolds
• An algorithmic approach to construct crystallizations of $3$-manifolds from presentations of fundamental groups
• A new family of triangulations of $\mathbb{R}{P}^d$