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The least spanning area of a knot and the optimal bounding chain problem
 In Proceedings of the 27th Annual ACM Symposium on Computational Geometry (SoCG’11). ACM
, 2011
"... Two fundamental objects in knot theory are the minimal genus surface and the least area surface bounded by a knot in a 3dimensional manifold. When the knot is embedded in a general 3manifold, the problems of finding these surfaces were shown to be NPcomplete and NPhard respectively. However, th ..."
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Cited by 15 (2 self)
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Two fundamental objects in knot theory are the minimal genus surface and the least area surface bounded by a knot in a 3dimensional manifold. When the knot is embedded in a general 3manifold, the problems of finding these surfaces were shown to be NPcomplete and NPhard respectively. However, there is evidence that the special case when the ambient manifold is R3, or more generally when the second homology is trivial, should be considerably more tractable. Indeed, we show here that a natural discrete version of the least area surface can be found in polynomial time. The precise setting is that the knot is a 1dimensional subcomplex of a triangulation of the ambient 3manifold. The main tool we use is a linear programming formulation of the Optimal Bounding Chain Problem (OBCP), where one is required to find the smallest norm chain with a given boundary. While the decision variant of OBCP is NPcomplete in general, we give conditions under which it can be solved in polynomial time. We then show that the least area surface can be constructed from the optimal bounding chain using a standard desingularization argument from 3dimensional topology. We also prove that the related Optimal Homologous Chain Problem is NPcomplete for homology with integer coefficients, complementing the corresponding result of Chen and Freedman for mod 2 homology.
Converting Between Quadrilateral and Standard Solution Sets in Normal Surface Theory
, 2009
"... The enumeration of normal surfaces is a crucial but very slow operation in algorithmic 3–manifold topology. At the heart of this operation is a polytope vertex enumeration in a highdimensional space (standard coordinates). Tollefson’s Q– theory speeds up this operation by using a much smaller space ..."
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The enumeration of normal surfaces is a crucial but very slow operation in algorithmic 3–manifold topology. At the heart of this operation is a polytope vertex enumeration in a highdimensional space (standard coordinates). Tollefson’s Q– theory speeds up this operation by using a much smaller space (quadrilateral coordinates), at the cost of a reduced solution set that might not always be sufficient for our needs. In this paper we present algorithms for converting between solution sets in quadrilateral and standard coordinates. As a consequence we obtain a new algorithm for enumerating all standard vertex normal surfaces, yielding both the speed of quadrilateral coordinates and the wider applicability of standard coordinates. Experimentation with the software package Regina shows this new algorithm to be extremely fast in practice, improving speed for large cases by factors from thousands up to millions.
Converting between quadrilateral and standard solution sets in
"... normal surface theory ..."
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THE UNKNOTTING PROBLEM AND
"... Abstract. Tollefson described a variant of normal surface theory for 3manifolds, called Qtheory, where only the quadrilateral coordinates are used. Suppose M is a triangulated, compact, irreducible, boundaryirreducible 3manifold. In Qtheory, if M contains an essential surface, then the projecti ..."
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Abstract. Tollefson described a variant of normal surface theory for 3manifolds, called Qtheory, where only the quadrilateral coordinates are used. Suppose M is a triangulated, compact, irreducible, boundaryirreducible 3manifold. In Qtheory, if M contains an essential surface, then the projective solution space has an essential surface at a vertex. One interesting situation not covered by this theorem is when M is boundary reducible, e.g. M is an unknot complement. We prove that in this case M has an essential disc at a vertex of the Qprojective solution space. 1.