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A random tunnel number one 3manifold does not fiber over the circle
 Geom. Topol
"... We address the question: how common is it for a 3–manifold to fiber over the circle? One motivation for considering this is to give insight into the fairly inscrutable Virtual Fibration Conjecture. For the special class of 3–manifolds with tunnel number one, we provide compelling theoretical and exp ..."
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We address the question: how common is it for a 3–manifold to fiber over the circle? One motivation for considering this is to give insight into the fairly inscrutable Virtual Fibration Conjecture. For the special class of 3–manifolds with tunnel number one, we provide compelling theoretical and experimental evidence that fibering is a very rare property. Indeed, in various precise senses it happens with probability 0. Our main theorem is that this is true for a measured lamination model of random tunnel number one 3–manifolds. The first ingredient is an algorithm of K Brown which can decide if a given tunnel number one 3–manifold fibers over the circle. Following the lead of Agol, Hass and W Thurston, we implement Brown’s algorithm very efficiently by working in the context of train tracks/interval exchanges. To analyze the resulting algorithm, we generalize work of Kerckhoff to understand the dynamics of splitting sequences of complete genus 2 interval exchanges. Combining all of this with a “magic splitting sequence ” and work of Mirzakhani proves the main theorem. The 3–manifold situation contrasts markedly with random 2–generator 1–relator groups; in particular, we show that such groups “fiber ” with probability strictly
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 8 (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.
PROBLEMS IN FOLIATIONS AND LAMINATIONS OF 3–MANIFOLDS
, 2002
"... 1.1. Notation. I will try to use consistent notation throughout, so for instance, F, G denote foliations, Λ denotes a lamination, λ, µ, ν denote leaves, etc. For a given object X in a manifold M, ˜ X will denote the pullback of X to the universal cover ˜ M. 1.2. Attribution. I have tried to credit ..."
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Cited by 2 (0 self)
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1.1. Notation. I will try to use consistent notation throughout, so for instance, F, G denote foliations, Λ denotes a lamination, λ, µ, ν denote leaves, etc. For a given object X in a manifold M, ˜ X will denote the pullback of X to the universal cover ˜ M. 1.2. Attribution. I have tried to credit questions to their originators. There are certain
NORMAL SURFACE THEORY IN LINK DIAGRAMS
, 2007
"... We give a diagrammatic variant of Haken’s normal surface theory, which relies only on a knot diagram and not on additional structures such as a triangulation. The Menasco–Thistlethwaite crossing bubble technique is used to represent surfaces as curve systems. These curves are represented by normal ..."
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We give a diagrammatic variant of Haken’s normal surface theory, which relies only on a knot diagram and not on additional structures such as a triangulation. The Menasco–Thistlethwaite crossing bubble technique is used to represent surfaces as curve systems. These curves are represented by normal arcs in regions of the diagram and integer linear equations are obtained by gluing the arcs in adjacent regions. We demonstrate an unknot recognition algorithm utilizing these techniques and give examples showing how the number of variables can be greatly reduced by diagrammatic constraints.
MENASCO NORMAL FORM AND RECOGNIZING UNKNOT DIAGRAMS
, 2007
"... We give a diagrammatic variant of Haken’s normal surface theory, which relies only on a knot diagram and not on additional structures such as a triangulation. The variables are normal arcs rather than normal discs. The crucial ingredient is Menasco’s crossing bubble technique. We demonstrate an unk ..."
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We give a diagrammatic variant of Haken’s normal surface theory, which relies only on a knot diagram and not on additional structures such as a triangulation. The variables are normal arcs rather than normal discs. The crucial ingredient is Menasco’s crossing bubble technique. We demonstrate an unknot recognition algorithm utilizing these techniques.
Linkless and flat embeddings in 3space and the Unknot problem (Extended Abstract)
 SCG'10
, 2010
"... ..."
NORMAL SURFACE THEORY IN LINK DIAGRAMS CHANHO SUH
"... Abstract. We give a diagrammatic variant of Haken’s normal surface theory, which relies only on a knot diagram and not on additional structures such as a triangulation. The Menasco–Thistlethwaite crossing bubble technique is used to represent surfaces as curve systems. These curves are represented b ..."
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Abstract. We give a diagrammatic variant of Haken’s normal surface theory, which relies only on a knot diagram and not on additional structures such as a triangulation. The Menasco–Thistlethwaite crossing bubble technique is used to represent surfaces as curve systems. These curves are represented by normal arcs in regions of the diagram and integer linear equations are obtained by gluing the arcs in adjacent regions. We demonstrate an unknot recognition algorithm utilizing these techniques and give examples showing how the number of variables can be greatly reduced by diagrammatic constraints. 1.
CROSSCAP NUMBERS AND THE JONES POLYNOMIAL
"... Abstract. We give sharp twosided linear bounds of the crosscap number (nonorientable genus) of alternating links in terms of their Jones polynomial. Our estimates are often exact and we use them to calculate the crosscap numbers for several infinite families of alternating links and for several a ..."
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Abstract. We give sharp twosided linear bounds of the crosscap number (nonorientable genus) of alternating links in terms of their Jones polynomial. Our estimates are often exact and we use them to calculate the crosscap numbers for several infinite families of alternating links and for several alternating knots with up to twelve crossings. We also discuss generalizations of our results for classes of nonalternating links. 1.
Hardness of embedding simplicial complexes in R^d
, 2009
"... Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into d? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for a ..."
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Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into d? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for all k ≥ 3. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5sphere implies that EMBEDd→d and EMBED (d−1)→d are undecidable for each d ≥ 5. Our main result is NPhardness of EMBED2→4 and, more generally, of EMBEDk→d for all k, d with d ≥ 4 and d ≥ k ≥ (2d −2)/3. These dimensions fall outside the metastable range of a theorem of Haefliger and Weber, which characterizes embeddability using the deleted product obstruction. Our reductions are based on examples, due to Segal, Spie˙z, Freedman, Krushkal, Teichner, and Skopenkov, showing that outside the metastable range the deleted product obstruction is not sufficient to characterize embeddability.