The size of spanning disks for polygonal curves

Let $K$ be a closed polygonal curve in \RR^3 consisting of $n$ line segments. Assume that $K$ is unknotted, so that it is the boundary of an embedded disk in \RR^3. This paper considers the question: How many triangles are needed to triangulate a Piecewise-Linear (PL) spanning disk of $K$? The main result exhibits a family of unknotted polygons with $n$ edges, $n \to \infty$, such that the minimal number of triangles needed in any triangulated spanning disk grows exponentially with $n$. For each integer $n \ge 0$, there is a closed, unknotted, polygonal curve $K_n$ in $R^3$ having less than $10n+9$ edges, with the property that any Piecewise-Linear triangulated disk spanning the curve contains at least $2^{n-1}$ triangles.Comment: 17 pages, 16 figure