4,973 research outputs found
How to make a triangulation of S^3 polytopal
We introduce a numerical isomorphism invariant p(T) for any triangulation T
of S^3. Although its definition is purely topological (inspired by the bridge
number of knots), p(T) reflects the geometric properties of T.
Specifically, if T is polytopal or shellable then p(T) is `small' in the
sense that we obtain a linear upper bound for p(T) in the number n=n(T) of
tetrahedra of T.
Conversely, if p(T) is `small' then T is `almost' polytopal, since we show
how to transform T into a polytopal triangulation by O((p(T))^2) local
subdivisions. The minimal number of local subdivisions needed to transform T
into a polytopal triangulation is at least .
Using our previous results [math.GT/0007032], we obtain a general upper bound
for p(T) exponential in n^2. We prove here by explicit constructions that there
is no general subexponential upper bound for p(T) in n. Thus, we obtain
triangulations that are `very far' from being polytopal.
Our results yield a recognition algorithm for S^3 that is conceptually
simpler, though somewhat slower, as the famous Rubinstein-Thompson algorithm.Comment: 24 pages, 17 figures. Final versio
Complexity of triangulations of the projective space
It is known that any two triangulations of a compact 3-manifold are related
by finite sequences of certain local transformations. We prove here an upper
bound for the length of a shortest transformation sequence relating any two
triangulations of the 3-dimensional projective space, in terms of the number of
tetrahedra.Comment: 10 pages, 3 figures. Revised version, to appear in Top. App
Persistent homology of groups
We introduce and investigate notions of persistent homology for p-groups and
for coclass trees of p-groups. Using computer techniques we show that
persistent homology provides fairly strong homological invariants for p-groups
of order at most 81. The strength of these invariants, and some elementary
theoretical properties, suggest that persistent homology may be a useful tool
in the study of prime-power groups.Comment: 12 pages, 6 figure
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Incremental closure for systems of two variables per inequality
Subclasses of linear inequalities where each inequality has at most two vari- ables are popular in abstract interpretation and model checking, because they strike a balance between what can be described and what can be efficiently computed. This paper focuses on the TVPI class of inequalities, for which each coefficient of each two variable inequality is unrestricted. An implied TVPI in- equality can be generated from a pair of TVPI inequalities by eliminating a given common variable (echoing resolution on clauses). This operation, called result , can be applied to derive TVPI inequalities which are entailed (implied) by a given TVPI system. The key operation on TVPI is calculating closure: satisfiability can be observed from a closed system and a closed system also simplifies the calculation of other operations. A closed system can be derived by repeatedly applying the result operator. The process of adding a single TVPI inequality to an already closed input TVPI system and then finding the closure of this augmented system is called incremental closure. This too can be calcu- lated by the repeated application of the result operator. This paper studies the calculus defined by result , the structure of result derivations, and how deriva- tions can be combined and controlled. A series of lemmata on derivations are presented that, collectively, provide a pathway for synthesising an algorithm for incremental closure. The complexity of the incremental closure algorithm is analysed and found to be O (( n 2 + m 2 )lg( m )), where n is the number of variables and m the number of inequalities of the input TVPI system
The computation of the cohomology rings of all groups of order 128
We describe the computation of the mod-2 cohomology rings of all 2328 groups
of order 128. One consequence is that all groups of order less than 256 satisfy
the strong form of Benson's Regularity Conjecture.Comment: 15 pages; revised versio
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