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Let $K$ be a closed polygonal curve in $\RR^3$ consisting of $n$ linesegments. Assume that $K$ is unknotted, so that it is the boundary of anembedded disk in $\RR^3$. This paper considers the question: How many trianglesare needed to triangulate a Piecewise-Linear (PL) spanning disk of $K$? Themain result exhibits a family of unknotted polygons with $n$ edges, $n \to\infty$, such that the minimal number of triangles needed in any triangulatedspanning disk grows exponentially with $n$. For each integer $n \ge 0$, thereis 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 spanningthe curve contains at least $2^{n-1}$ triangles.

The Size of Spanning Disks for Polygonal Curves