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The minimal number of triangles needed to span a polygon embedded in R d
 J. GoodmanR. Pollack Festscrift Volume
, 2003
"... Given a closed polygon P having n edges, embedded in R d, we give upper and lower bounds for the minimal number of triangles t needed to form a triangulated PL surface embedded in R d having P as its geometric boundary. More generally we obtain such bounds for a triangulated (locally flat) PL surfac ..."
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Given a closed polygon P having n edges, embedded in R d, we give upper and lower bounds for the minimal number of triangles t needed to form a triangulated PL surface embedded in R d having P as its geometric boundary. More generally we obtain such bounds for a triangulated (locally flat) PL surface having P as its boundary which is immersed in R d and whose interior is disjoint from P. The most interesting case is dimension 3, where the polygon may be knotted. We use the Seifert surface construction to show that for any polygon embedded in R 3 there exists an embedded orientable triangulated PL surface having at most 7n 2 triangles, whose boundary is a subdivision of P. We complement this with a construction of families of polygons with n vertices for which any such embedded surface requires at least 1 2 n2 − O(n) triangles. We also exhibit families of polygons in R 3 for which Ω(n 2) triangles are required in any immersed PL surface of the above kind. In contrast, in dimension 2 and in dimensions d ≥ 5 there always exists an embedded locally flat PL disk having P as boundary that contains at most n triangles. In dimension 4 there always exists an immersed locally flat PL disk of the above kind that contains at most 3n triangles. An unresolved case is that of embedded PL surfaces in dimension 4, where we establish only an O(n 2) upper bound. These results can be viewed as providing qualitative discrete analogues of the isoperimetric inequality for piecewise linear (PL) manifolds. In dimension 3 they imply that the (asymptotic) discrete isoperimetric constant lies between 1/2 and 7. Keywords: isoperimetric inequality, Plateau’s problem, computational complexity
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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
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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
EMBEDDED PLATEAU PROBLEM
"... Abstract. We show that if Γ is a simple closed curve bounding an embedded disk in a closed 3manifold M, then there exists a disk Σ in M with boundary Γ such that Σ minimizes the area among the embedded disks with boundary Γ. Moreover, Σ is smooth, minimal and embedded everywhere except where the bo ..."
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Abstract. We show that if Γ is a simple closed curve bounding an embedded disk in a closed 3manifold M, then there exists a disk Σ in M with boundary Γ such that Σ minimizes the area among the embedded disks with boundary Γ. Moreover, Σ is smooth, minimal and embedded everywhere except where the boundary Γ meets the interior of Σ. The same result is also valid for homogeneously regular manifolds with sufficiently convex boundary. 1.
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.