13 papers
Area Inequalities for Embedded Disks Spanning Unknotted Curves
Joel Hass, Jeffrey C. Lagarias, William P. Thurston
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 lengt…
The Number of Triangles Needed to Span a Polygon Embedded in R^d
Joel Hass, Jeffrey C. Lagarias
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 in R^d havin…
The Computational Complexity of Knot Genus and Spanning Area
Ian Agol, Joel Hass, William P. Thurston
We investigate the computational complexity of some problems in three-dimensional topology and geometry. We show that the problem of determining a bound on the genus of a knot in a…
Affine isoperimetric inequalities for piecewise linear surfaces
Joel Hass, Jeffrey C. Lagarias
This paper considers affine analogues of the isoperimetric inequality in the sense of piecewise linear topology. Given a closed polygon P embedded in R^d having n edges, we give up…
Double Bubbles Minimize
Joel Hass, Roger Schlafly
The classical isoperimetric inequality in R^3 states that the surface of smallest area enclosing a given volume is a sphere. We show that the least area surface enclosing two equal…
On finiteness of the number of boundary slopes of immersed surfaces in 3-manifolds
Joel Hass, Shicheng Wang, Qing Zhou
For any hyperbolic 3-manifold with totally geodesic boundary, there are finitely many boundary slopes for essential immersed surfaces of a given genus. There is a uniform bound…