12 research outputs found

    A note about complexity of lens spaces

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    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.Comment: 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 essentia

    2-symmetric transformations for 3-manifolds of genus two

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    As previously known, all 3-manifolds of genus two can be represented by edge-coloured graphs uniquely defined by 6-tuples of integers satisfying simple conditions. The present paper describes an ``elementary transformation'' on these 6-tuples which changes the associated graph but does not change the represented manifold. This operation is a useful tool in the classification problem for 3-manifolds of genus two; in fact, it allows to define an equivalence relation on ``admissible'' 6-tuples so that equivalent 6-tuples represent the same manifold. Different equivalence classes can represent the same manifold; however, equivalence classes ``almost always'' contain infinitely many 6-tuples. Finally, minimal representatives of the equivalence classes are described.Comment: 27 pages, 10 figures, to appear on Journal of Combinatorial Theory Series

    A note on irreducible Heegaard diagrams

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    AbstractWe construct a Heegaard diagram of genus three for the real projective 3-space, which has no waves and pairs of complementary handles. The first example was given by Im and Kim but our diagram has smaller complexity. Furthermore the proof presented here is quite different to that of the quoted authors, and permits also to obtain a simple alternative proof of their result. Examples of irreducible Heegaard diagrams of certain connected sums complete the paper.We construct a Heegaard diagram of genus three for the real projective 3-space, which has no waves and pairs of complementary handles. The first example was given by Im and Kim but our diagram has smaller complexity. Furthermore the proof presented here is quite different to that of the quoted authors, and permits also to obtain a simple alternative proof of their result. Examples of irreducible Heegaard diagrams of certain connected sums complete the paper

    A note on some homology spheres which are 2-fold coverings of inequivalent knots

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    We construct a family of closed 3--manifolds Mα,rM_{\alpha,r}, which are  homeomorphic to the Brieskorn homology spheres Σ(2,α+1,q+2r−1)\Sigma(2, \alpha+1, q+2r-1),  where q=α(r−1)q=\alpha(r-1) and both α≥1\alpha \ge 1 and q≥3q \ge 3 are odd. We show  that Mα,rM_{\alpha,r} can be represented as 2--fold covering of the 3--sphere  branched over two inequivalent knots. Our proofs follow immediately from two  different symmetries of a genus 2 Heegaard diagram of Σ(2,α+1,  q+2r−1)\Sigma(2, \alpha+1,  q+2r-1), and generalize analogous results proved in [BGM], [IK], [SIK] and  [T]

    A census of genus two 3-manifolds up to 42 coloured tetrahedra

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    We improve and extend to the non-orientable case a recent result of Karabas, Malicki and Nedela concerning the classification of all orientable prime 3-manifolds of Heegaard genus two, triangulated with at most 42 coloured tetrahedra.Comment: 24 pages, 3 figure

    Triangulations

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    The earliest work in topology was often based on explicit combinatorial models – usually triangulations – for the spaces being studied. Although algebraic methods in topology gradually replaced combinatorial ones in the mid-1900s, the emergence of computers later revitalized the study of triangulations. By now there are several distinct mathematical communities actively doing work on different aspects of triangulations. The goal of this workshop was to bring the researchers from these various communities together to stimulate interaction and to benefit from the exchange of ideas and methods

    Author index to volume 87 (1991)

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