摘要

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 >= 3, 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 recently been 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 >= 3, the following relation: k(L(p, q)) = 5 + 2c(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.

  • 出版日期2015-11