Isoperimetric inequalities for unknotting disks
We establish new isoperimetric inequalities for smooth unknotted curves
in R3. The area of an embedded disk spanning such
a curve is bounded above in terms of two parameters: the length L
of the curve and its thickness r, the maximal radius of an embedded
tubular neighborhood around the curve. For curves of fixed length,
the expression giving the upper bound on the area grows exponentially in
1/r2. In the direction of lower bounds, we give
a sequence of length one curves with r 
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 A0, there is a smooth, unknotted curve of length
one for which the area of a smallest embedded spanning disk is greater
than A0.
This is joint work with Jeff Lagarias and Bill Thurston.
Fall
2002 PNGS meeting