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Area inequalities for embedded disks spanning unknotted curves
 2003, arXiv:math.DG/0306313. EFFICIENTLY BOUND 4MANIFOLDS 43
"... We show that a smooth unknotted curve in R 3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length ..."
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We show that a smooth unknotted curve in R 3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length, the expression giving the upper bound on the area grows exponentially in 1/r 2. In the direction of lower bounds, we give a sequence of length one curves with r→0for which the area of any spanning disk is bounded from below by a function that grows exponentially with 1/r. In particular, given any constant A, there is a smooth, unknotted length one curve for which the area of a smallest embedded spanning disk is greater than A. 1
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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, 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 ..."
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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
3manifolds efficiently bound 4manifolds
"... Costantino and Dylan Thurston It has been 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 little attention has been paid to the complexity of the 4manifold produced. Given a 3manifold ..."
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Costantino and Dylan Thurston It has been 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 little attention has been paid 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(n2), 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, we have c1G S c2G2, for suitable constants c1, c2. In particular, the manifolds with shadow complexity 0 are the graph manifolds. In addition, we give an O(n4) 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 Gromov norm G to R2 has at least G/10 crossing singularities, and if M is hyperbolic there is a map with at most c3G2 crossing singularities. 1.