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3–manifolds efficiently bound 4–manifolds
, 2005
"... Abstract. It is known since 1954 that every 3manifold bounds a 4manifold. Thus, for instance, every 3manifold has a surgery diagram. There are several proofs of this fact, but there has been little attention to the complexity of the 4manifold produced. Given a 3manifold M 3 of complexity n, we ..."
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Cited by 13 (5 self)
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Abstract. It is known since 1954 that every 3manifold bounds a 4manifold. Thus, for instance, every 3manifold has a surgery diagram. There are several proofs of this fact, but there has been little attention to the complexity of the 4manifold produced. Given a 3manifold M 3 of complexity n, we construct a 4manifold bounded by M of complexity O(n 2), where the “complexity ” of a piecewiselinear manifold is the minimum number of nsimplices in a triangulation. The proof goes through the notion of “shadow complexity ” of a 3manifold M. A shadow of M is a wellbehaved 2dimensional spine of a 4manifold bounded by M. We further prove that, for a manifold M satisfying the Geometrization Conjecture with Gromov norm G and shadow complexity S, c1G ≤ S ≤ c2G 2 for suitable constants c1, c2. In particular, the manifolds with shadow complexity 0 are the graph manifolds. In addition, we give an O(n 4) bound for the complexity of a spin 4manifold bounding a given spin 3manifold. We also show that every stable map from a 3manifold M with
Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle
, 2007
"... We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in S 2 and in R 2. 1 ..."
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Cited by 4 (1 self)
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We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in S 2 and in R 2. 1
Spiralling and Folding: The Topological View
"... For every n, we construct two arcs in the fourpunctured sphere that have at least n intersections and which do not form spirals. This is accomplished in several steps: we first exhibit closed curves on the torus that do not form double spirals, then arcs on the fourpunctured torus that do not form ..."
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Cited by 1 (1 self)
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For every n, we construct two arcs in the fourpunctured sphere that have at least n intersections and which do not form spirals. This is accomplished in several steps: we first exhibit closed curves on the torus that do not form double spirals, then arcs on the fourpunctured torus that do not form spirals, and finally arcs in the fourpunctured sphere which do not form spirals. string graphs whose drawings require Ω(2 n) intersections. This figure also indicates a bigon, but it can easily be eliminated by puncturing the surface or adding genus in the middle of the bigon. So while a fold without a bigon does not yield an obvious simplification, it does indicate topological complexity of the surface. 1
NONORIENTABLE FUNDAMENTAL SURFACES IN LENS SPACES
, 809
"... Abstract. We give a concrete example of an infinite sequence of (pn, qn)lens spaces L(pn, qn) with natural triangulations T(pn, qn) with pn taterahedra such that L(pn, qn) contains a certain nonorientable closed surface which is fundamental with respect to T(pn, qn) and of minimal crosscap number ..."
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Cited by 1 (1 self)
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Abstract. We give a concrete example of an infinite sequence of (pn, qn)lens spaces L(pn, qn) with natural triangulations T(pn, qn) with pn taterahedra such that L(pn, qn) contains a certain nonorientable closed surface which is fundamental with respect to T(pn, qn) and of minimal crosscap number among all closed nonorientable surfaces in L(pn, qn) and has n −2 parallel sheets of normal disks of a quadrilateral type disjoint from the pair of core circles of L(pn, qn). Actually, we can set p0 = 0, q0 = 1, pk+1 = 3pk + 2qk and qk+1 = pk + qk.
Shortest NonCrossing Walks in the Plane
, 2010
"... Let G be a plane graph with nonnegative edge weights, and let k terminal pairs be specified on h face boundaries. We present an algorithm to find k noncrossing walks in G of minimum total length that connect all terminal pairs, if any such walks exist, in 2 O(h2) n log k time. The computed walks m ..."
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Cited by 1 (0 self)
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Let G be a plane graph with nonnegative edge weights, and let k terminal pairs be specified on h face boundaries. We present an algorithm to find k noncrossing walks in G of minimum total length that connect all terminal pairs, if any such walks exist, in 2 O(h2) n log k time. The computed walks may overlap but may not cross each other or themselves. Our algorithm generalizes a result of Takahashi, Suzuki, and Nikizeki [Algorithmica 1996] for the special case h ≤ 2. We also describe an algorithm for the corresponding geometric problem, where the terminal points lie on the boundary of h polygonal obstacles of total complexity n, again in 2 O(h2) n time, generalizing an algorithm of Papadopoulou [Int. J. Comput. Geom. Appl. 1999] for the special case h ≤ 2. In both settings, shortest noncrossing walks can have complexity exponential in h. We also describe algorithms to determine in O(n) time whether the terminal pairs can be connected by any noncrossing walks.
unknown title
"... Among the different ways to combinatorially represent 3manifolds, two of the most popular are triangulations and surgery on a link. A triangulation is very natural way to represent 3manifolds, and any other representation of a 3manifold is easy to turn into a triangulation. On the other hand, alt ..."
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Among the different ways to combinatorially represent 3manifolds, two of the most popular are triangulations and surgery on a link. A triangulation is very natural way to represent 3manifolds, and any other representation of a 3manifold is easy to turn into a triangulation. On the other hand, although some 3manifold invariants may be
unknown title
"... Among the different ways to combinatorially represent 3manifolds, two of the most popular are triangulations and surgery on a link. A triangulation is very natural way to represent 3manifolds, and any other representation of a 3manifold is easy to turn into a triangulation. On the other hand, alt ..."
Abstract
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Among the different ways to combinatorially represent 3manifolds, two of the most popular are triangulations and surgery on a link. A triangulation is very natural way to represent 3manifolds, and any other representation of a 3manifold is easy to turn into a triangulation. On the other hand, although some 3manifold invariants may be
IEEE INTERNATIONAL CONFERENCE ON SHAPE MODELING AND APPLICATIONS (SMI) 2009 1 Filling Holes in Triangular Meshes by Curve Unfolding
"... Abstract—We propose a novel approach to automatically fill holes in triangulated models. Each hole is filled using a minimum energy surface that is obtained in three steps. First, we unfold the hole boundary onto a plane using energy minimization. Second, we triangulate the unfolded hole using a con ..."
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Abstract—We propose a novel approach to automatically fill holes in triangulated models. Each hole is filled using a minimum energy surface that is obtained in three steps. First, we unfold the hole boundary onto a plane using energy minimization. Second, we triangulate the unfolded hole using a constrained Delaunay triangulation. Third, we embed the triangular mesh as a minimum energy surface in R 3. The running time of the method depends primarily on the size of the hole boundary and not on the size of the model, thereby making the method applicable to large models. Our experiments demonstrate the applicability of the algorithm to the problem of filling holes bounded by highly curved boundaries in large models.
hole˙filling˙journal Filling Holes in Triangular Meshes Using Digital Images by Curve Unfolding ∗
"... We propose a novel approach to automatically fill holes in triangulated models. Each hole is filled using a minimum energy surface that is obtained in three steps. First, we unfold the hole boundary onto a plane using energy minimization. Second, we triangulate the unfolded hole using a constrained ..."
Abstract
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We propose a novel approach to automatically fill holes in triangulated models. Each hole is filled using a minimum energy surface that is obtained in three steps. First, we unfold the hole boundary onto a plane using energy minimization. Second, we triangulate the unfolded hole using a constrained Delaunay triangulation. Third, we embed the triangular mesh as a minimum energy surface in R 3. When embedding the triangular mesh, any energy function can be used to estimate the missing data. We use a variational multiview approach to estimate the missing data. The running time of the method depends primarily on the size of the hole boundary and not on the size of the model, thereby making