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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
How to make a triangulation of S 3 polytopal
 Trans. Am. Math. Soc
, 2004
"... We introduce a method of studying triangulations T of S 3 that connects topological and geometrical approaches. In its center is a new numerical invariant p(T), called polytopality. Although its definition is purely topological, it is sensitive for geometric properties of T, as follows. We consider ..."
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We introduce a method of studying triangulations T of S 3 that connects topological and geometrical approaches. In its center is a new numerical invariant p(T), called polytopality. Although its definition is purely topological, it is sensitive for geometric properties of T, as follows. We consider local moves called expansions, that generalize stellar subdivisions of simplicial complexes. Let d(T) be the length of a shortest sequence of expansions relating T with the boundary complex of a convex 4–polytope. In this paper we obtain both lower and upper bounds for d(T) in terms of p(T). Using previous results [9] based on the Rubinstein–Thompson algorithm, we obtain an upper bound for d(T) in terms of the number n of tetrahedra of T. The bound is exponential in n 2, and we prove here that in general one can not replace it by a subexponential bound. Our results yield another recognition algorithm for S 3 that is conceptionally much simpler, though slower, as the Rubinstein–Thompson algorithm.
Complexity of triangulations of the projective space
"... It is known that any two triangulations of a compact 3–manifold are related by finite sequences of certain local transformations. We prove here an upper bound for the length of a shortest transformation sequence relating any two triangulations of the 3–dimensional projective space, in terms of the n ..."
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It is known that any two triangulations of a compact 3–manifold are related by finite sequences of certain local transformations. We prove here an upper bound for the length of a shortest transformation sequence relating any two triangulations of the 3–dimensional projective space, in terms of the number of tetrahedra.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
, 2004
"... Abstract. We introduce a numerical isomorphism invariant p(T) for any triangulation T of S3. Although its definition is purely topological (inspired by the bridge number of knots), p(T) reflects the geometric properties of T. Specifically, if T is polytopal or shellable then p(T) is “small ” in the ..."
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Abstract. We introduce a numerical isomorphism invariant p(T) for any triangulation T of S3. Although its definition is purely topological (inspired by the bridge number of knots), p(T) reflects the geometric properties of T. Specifically, if T is polytopal or shellable then p(T) is “small ” in the sense that we obtain a linear upper bound for p(T) in the number n = n(T) of tetrahedra of T. Conversely, if p(T) is “small ” then T is “almost ” polytopal, since we show how to transform T into a polytopal triangulation by O((p(T)) 2) local subdivisions. The minimal number of local subdivisions needed to transform p(T) T into a polytopal triangulation is at least − n − 2. 3n Using our previous results [The size of triangulations supporting a given link. Geometry & Topology 5 (2001), 369–398], we obtain a general upper bound for p(T) exponential in n2. We prove here by explicit constructions that there is no general subexponential upper bound for p(T) in n. Thus, we obtain triangulations that are “very far ” from being polytopal. Our results yield a recognition algorithm for S3 that is conceptually simpler, though somewhat slower, as the famous Rubinstein–Thompson algorithm. 1.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 356, Number 11, Pages 4519–4542
, 2004
"... Abstract. We introduce a numerical isomorphism invariant p(T) for any triangulation T of S3. Although its definition is purely topological (inspired by the bridge number of knots), p(T) reflects the geometric properties of T. Specifically, if T is polytopal or shellable, then p(T) is “small ” in the ..."
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Abstract. We introduce a numerical isomorphism invariant p(T) for any triangulation T of S3. Although its definition is purely topological (inspired by the bridge number of knots), p(T) reflects the geometric properties of T. Specifically, if T is polytopal or shellable, then p(T) is “small ” in the sense that we obtain a linear upper bound for p(T)inthenumbern = n(T) of tetrahedra of T. Conversely, if p(T) is “small”, then T is “almost ” polytopal, since we show how to transform T into a polytopal triangulation by O((p(T)) 2)local subdivisions. The minimal number of local subdivisions needed to transform p(T) T into a polytopal triangulation is at least − n − 2. 3n Using our previous results [The size of triangulations supporting a given link, Geometry & Topology 5 (2001), 369–398], we obtain a general upper bound for p(T) exponential in n2. We prove here by explicit constructions that there is no general subexponential upper bound for p(T)inn. Thus,we obtain triangulations that are “very far ” from being polytopal. Our results yield a recognition algorithm for S3 that is conceptually simpler, although somewhat slower, than the famous Rubinstein–Thompson algorithm. 1.
HOW TO MAKE A TRIANGULATION OF S 3
, 2004
"... Abstract. We introduce a numerical isomorphism invariant p(T) for any triangulation T of S3. Although its definition is purely topological (inspired by the bridge number of knots), p(T) reflects the geometric properties of T. Specifically, if T is polytopal or shellable, then p(T) is “small ” in the ..."
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Abstract. We introduce a numerical isomorphism invariant p(T) for any triangulation T of S3. Although its definition is purely topological (inspired by the bridge number of knots), p(T) reflects the geometric properties of T. Specifically, if T is polytopal or shellable, then p(T) is “small ” in the sense that we obtain a linear upper bound for p(T)inthenumbern = n(T) of tetrahedra of T. Conversely, if p(T) is “small”, then T is “almost ” polytopal, since we show how to transform T into a polytopal triangulation by O((p(T)) 2)local subdivisions. The minimal number of local subdivisions needed to transform p(T) T into a polytopal triangulation is at least − n − 2. 3n Using our previous results [The size of triangulations supporting a given link, Geometry & Topology 5 (2001), 369–398], we obtain a general upper bound for p(T) exponential in n2. We prove here by explicit constructions that there is no general subexponential upper bound for p(T)inn. Thus,we obtain triangulations that are “very far ” from being polytopal. Our results yield a recognition algorithm for S3 that is conceptually simpler, although somewhat slower, than the famous Rubinstein–Thompson algorithm. 1.