3,831 research outputs found
An algorithm to determine the Heegaard genus of simple 3-manifolds with non-empty boundary
We provide an algorithm to determine the Heegaard genus of simple 3-manifolds
with non-empty boundary. More generally, we supply an algorithm to determine
(up to ambient isotopy) all the Heegaard splittings of any given genus for the
manifold. As a consequence, the tunnel number of a hyperbolic link is
algorithmically computable. Our techniques rely on Rubinstein's work on almost
normal surfaces, and also a new structure called a partially flat angled ideal
triangulation.Comment: 23 pages, 14 figure
Bounds for the genus of a normal surface
This paper gives sharp linear bounds on the genus of a normal surface in a
triangulated compact, orientable 3--manifold in terms of the quadrilaterals in
its cell decomposition---different bounds arise from varying hypotheses on the
surface or triangulation. Two applications of these bounds are given. First,
the minimal triangulations of the product of a closed surface and the closed
interval are determined. Second, an alternative approach to the realisation
problem using normal surface theory is shown to be less powerful than its dual
method using subcomplexes of polytopes.Comment: 38 pages, 25 figure
The homeomorphism problem for closed 3-manifolds
We give a more geometric approach to an algorithm for deciding whether two
hyperbolic 3-manifolds are homeomorphic. We also give a more algebraic approach
to the homeomorphism problem for geometric, but non-hyperbolic, 3-manifolds.Comment: first version: 12 pages. Replacement: 14 pages. Includes minor
improvements to exposition in response to referee's comment
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