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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
Linkless and flat embeddings in 3space and the Unknot problem (Extended Abstract)
 SCG'10
, 2010
"... ..."
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.
NORMAL SURFACE THEORY IN LINK DIAGRAMS
, 708
"... 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.
MENASCO NORMAL FORM AND RECOGNIZING UNKNOT DIAGRAMS
, 708
"... 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 variables are normal arcs rather than normal discs. The crucial ingredient is Menasco’s crossing bubble technique. We demonstra ..."
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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 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. 1.