Area Inequalities for Embedded Disks Spanning Unknotted Curves

Author(s): Hass, Joel; Lagarias, Jeffrey C.; Thurston, William P. | 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 approaching 0 for 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]  F. Almgren,et al.  Optimal isoperimetric inequalities , 1985 .

[2]  F. Morgan.  Geometric Measure Theory: A Beginner's Guide , 1988 .

[3]  Jeffrey C. Lagarias,et al.  The computational complexity of knot and link problems , 1999, JACM.

[4]  W. Reid.  The Isoperimetric Inequality and Associated Boundary Problems , 1959 .

[5]  T. O’Neil.  Geometric Measure Theory , 2002 .

[6]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[7]  R. Courant.  On the Problem of Plateau. , 1936, Proceedings of the National Academy of Sciences of the United States of America.

[8]  B. Dundas,et al.  DIFFERENTIAL TOPOLOGY , 2002 .

[9]  Eric J. Rawdon,et al.  Thickness of knots , 1999 .

[10]  J. Douglas.  Solution of the problem of Plateau , 1931 .

[11]  Bernard Chazelle,et al.  Convex Partitions of Polyhedra: A Lower Bound and Worst-Case Optimal Algorithm , 1984, SIAM J. Comput..

[12]  J. Lagarias,et al.  The Minimal Number of Triangles Needed to Span a Polygon Embedded in ℝd , 2003 .

[13]  Jeffrey C. Lagarias,et al.  The computational complexity of knot and link problems , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[14]  G. Buck,et al.  Thickness and crossing number of knots , 1999 .

[15]  L. Bianchi.  Vorlesungen über Differentialgeometrie I , 1899 .

[17]  Robert Gulliver,et al.  Regularity of Minimizing Surfaces of Prescribed Mean Curvature , 1973 .

[18]  E. Beckenbach,et al.  Subharmonic functions and surfaces of negative curvature , 1933 .

[19]  Robert Osserman,et al.  A Proof of the Regularity Everywhere of the Classical Solution to Plateau's Problem , 1970 .

[20]  Jason H. Cantarella,et al.  On the minimum ropelength of knots and links , 2001, math/0103224.

[21]  R. Osserman.  The isoperimetric inequality , 1978 .

[22]  David Avis,et al.  Triangulating point sets in space , 1987, Discret. Comput. Geom..

[23]  J. Lagarias,et al.  The number of Reidemeister moves needed for unknotting , 1998, math/9807012.