176 research outputs found

    Minimal flag triangulations of lower-dimensional manifolds

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    We prove the following results on flag triangulations of 2- and 3-manifolds. In dimension 2, we prove that the vertex-minimal flag triangulations of RP2\mathbb{R} P^2 and S1×S1\mathbb{S}^1\times \mathbb{S}^1 have 11 and 12 vertices, respectively. In general, we show that 8+3k8+3k (resp. 8+4k8+4k) vertices suffice to obtain a flag triangulation of the connected sum of kk copies of RP2\mathbb{R} P^2 (resp. S1×S1\mathbb{S}^1\times \mathbb{S}^1). In dimension 3, we describe an algorithm based on the Lutz-Nevo theorem which provides supporting computational evidence for the following generalization of the Charney-Davis conjecture: for any flag 3-manifold, γ2:=f1−5f0+16≥16β1\gamma_2:=f_1-5f_0+16\geq 16 \beta_1, where fif_i is the number of ii-dimensional faces and β1\beta_1 is the first Betti number over a field. The conjecture is tight in the sense that for any value of β1\beta_1, there exists a flag 3-manifold for which the equality holds.Comment: 6 figures, 3 tables, 19 pages. Final version with a few typos correcte

    Balanced triangulations on few vertices and an implementation of cross-fips

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    A d -dimensional simplicial complex is balanced if the underlying graph is ( d + 1 ) -colorable. We present an implementation of cross-flips, a set of local moves introduced by Izmestiev, Klee and Novik which connect any two PL-homeomorphic balanced combinatorial manifolds. As a result we exhibit a vertex minimal balanced triangulation of the real projective plane, of the dunce hat and of the real projective space, as well as several balanced triangulations of surfaces and 3-manifolds on few vertices. In particular we construct small balanced triangulations of the 3-sphere that are non-shellable and shellable but not vertex decomposable

    Detecting genus in vertex links for the fast enumeration of 3-manifold triangulations

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    Enumerating all 3-manifold triangulations of a given size is a difficult but increasingly important problem in computational topology. A key difficulty for enumeration algorithms is that most combinatorial triangulations must be discarded because they do not represent topological 3-manifolds. In this paper we show how to preempt bad triangulations by detecting genus in partially-constructed vertex links, allowing us to prune the enumeration tree substantially. The key idea is to manipulate the boundary edges surrounding partial vertex links using expected logarithmic time operations. Practical testing shows the resulting enumeration algorithm to be significantly faster, with up to 249x speed-ups even for small problems where comparisons are feasible. We also discuss parallelisation, and describe new data sets that have been obtained using high-performance computing facilities.Comment: 16 pages, 7 figures, 3 tables; v2: minor revisions; to appear in ISSAC 201
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