## Area inequalities for embedded disks spanning unknotted curves

Venue: | 2003, arXiv:math.DG/0306313. EFFICIENTLY BOUND 4-MANIFOLDS 43 |

Citations: | 5 - 1 self |

### BibTeX

@INPROCEEDINGS{Hass_areainequalities,

author = {Joel Hass and Jeffrey C. Lagarias and William P. Thurston},

title = {Area inequalities for embedded disks spanning unknotted curves},

booktitle = {2003, arXiv:math.DG/0306313. EFFICIENTLY BOUND 4-MANIFOLDS 43},

year = {}

}

### OpenURL

### Abstract

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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Citation Context ... know whether such an improvement to O(n) is possible in the case above for general position vertices, because we impose the stronger requirement that P be in the 1-skeleton of the triangulation. See =-=[9]-=- for a possible approach. 8Proof of Theorem 2.1: (1) Let B be the triangulated ball constructed in Lemma 2.3, that has at most t = 290n2 +290n+116 tetrahedra. By hypothesis P is unknotted in R3 , and... |

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Citation Context ...y if and only if the curve is a circle. There are many extensions of the isoperimetric theorem to higher dimensions; some of these are discussed in Almgren [1], Burago and Zalgaller [7], and Osserman =-=[24]-=-. There are two natural extensions of the isoperimetric inequality to a simple closed C 2 -curve γ in R 3 of length L. (1) There exists an immersed smooth disk in R 3 , with γ as boundary, having area... |

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Citation Context ...red at the origin. By rotating 6the coordinate system if necessary, we can suppose that the projection of P in the z-direction onto the xy-plane is a regular projection, and then, as in Lemma 7.1 of =-=[17]-=-, this projection has less than n 2 crossings. The graph associated to the projection has straight-line edges, and its vertices consist of the projections of vertices of the polygon γ together with al... |

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Citation Context ... (2) Such an embedded PL disk can be chosen to lie inside a ball of radius 4L, and its area then satisfies A ≤ (C4) n2 L 2 . Theorem 2.1 is proved by a modification of ideas used in Hass and Lagarias =-=[15]-=-. That paper obtains upper bounds on the number of elementary moves required to move from an unknotted polygonal curve in the one-skeleton of a triangulated 3-manifold having t tetrahedra, to a single... |

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Citation Context ...in R 3 This appendix establishes two versions of the isoperimetric theorem in R 3 as stated as (1) and (2) in §1. Result (1) follows from the solution of the Plateau problem, due to Douglas and Rado, =-=[11]-=-, [25], and on a result of Reid [26]. We deduce (2) from (1), using standard cut and paste methods of 3-manifold topology, which can be used to convert an immersed surface to an embedded surface, poss... |

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Citation Context ... of Theorem 6.1: Span γ by a least area disk D, whose existence is guaranteed for rectifiable γ by the solution of the Plateau problem [11], [25]. The regularity results of Osserman [23] and Gulliver =-=[14]-=- show that D is a smooth 22immersion in its interior. (The interior of D may transversely intersect γ, as indeed it must if γ is knotted.) Reid [26] shows that 4πA ≤ L 2 , with equality if and only i... |

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Citation Context ...ality holds. Proof of Theorem 6.1: Span γ by a least area disk D, whose existence is guaranteed for rectifiable γ by the solution of the Plateau problem [12], [26]. The regularity results of Osserman =-=[24]-=- and Gulliver [15] show that D is a smooth immersion in its interior. (The interior of D may transversely intersect γ, as indeed it must if γ is knotted.) The argument in Reid [27] (see also [9]) show... |

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Citation Context ... it states that 4πA ≤ L 2 , (1) with equality if and only if the curve is a circle. There are many extensions of the isoperimetric theorem to higher dimensions; some of these are discussed in Almgren =-=[1]-=-, Burago and Zalgaller [7], and Osserman [24]. There are two natural extensions of the isoperimetric inequality to a simple closed C 2 -curve γ in R 3 of length L. (1) There exists an immersed smooth ... |

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Citation Context ...not type, called the minimal ropelength, is the inverse of the thickness. Various properties of ropelength, including the existence of a minimizer in the class C 1,1 , have been investigated recently =-=[8]-=-. Our first two results concern area bounds in terms of the length L and thickness r of an unknotted closed curve γ. In §3 we show that for an unknotted curve there is an upper bound for the area of a... |

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Citation Context ... A, with 4πA ≤ L 2 . If the curve is not a circle, then there exists such a surface for which strict inequality holds. Result (1) traces back to a 1933 isoperimetric inequality of Beckenbach and Rado =-=[4]-=- for surfaces of non-positive Gaussian curvature. Result (2) traces back to a 1930 argument in Blaschke [5, p. 247], see Osserman [24, p. 1202]. For completeness we indicate proofs of these well-known... |

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Citation Context ...Theorem 2.1 in §2. To deduce Theorem 1.1 from it, we show that we can obtain such a triangulated surface in which the area of each triangle is bounded above by a constant. Hass, Snoeyink and Thurston =-=[18]-=- established the following PL analogue of Theorems 1.2 and 1.3. Theorem 1.5. There is a constant C3 > 1 and an infinite sequence of unknotted polygonal curves {Kn}n=1,...,∞ having between n and 23n se... |

4 | The minimal number of triangles needed to span a polygon embedded
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Citation Context ...qualities, in which the “length” is replaced by the number of edges n in a polygon, and the “area” is the number of triangles in an embedded triangulated surface having the polygon as boundary, as in =-=[16]-=-. We establish the following PL analogue of Theorem 1.1 above. Theorem 1.4. There is a constant C2 > 1 such that given any polygonal unknotted curve P in R 3 containing n line segments, there is a tri... |

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2 |
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Citation Context ... obtain the desired triangulation, with B inside a sphere of radius 4L. Remark. We do not know whether the bound O(n 2 ) in the triangulation above is the optimal order of magnitude. Avis and ElGindy =-=[3]-=- give examples of sets of n points in R 3 , for arbitrarily large n, such that any triangulation of the convex hull including these points as vertices required at least Ω(n 2 ) tetrahedra. However, th... |

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Citation Context ...used to define a knot invariant. The supremum of the ratio r(γ)/L(γ) over all C 2 -curves γ having a given knot type defines a knot invariant called the thickness of a knot, studied in Buck and Simon =-=[6]-=- and Litherland et al. [21], where it is related to various knot energies. For a C 2 -closed curve γ the quantity L(γ)/r(γ) is called the ropelength of γ, and the infimum of the ropelength over all C ... |

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