The size of spanning disks for polygonal curves. (English) Zbl 1015.57008

Hass, Joel; Snoeyink, Jack; Thurston, William P.

Discrete Comput. Geom. 29, No. 1, 1-17 (2003).

Several algorithms devised in topology (to test knot triviality) or group theory (for the word problem) are based on a search for embedded piecewise-linear spanning disks. The main result of the paper under review shows that the disks can be exponentially more complicated than the boundary curves: For any positive integer \(n\), there exists an unknotted polygon \(K_n\) in \(\mathbb R ^3\) with at most \(10n+9\) edges such that any piecewise-smooth embedded disk spanning \(K_n\) intersects the y-axis in at least \(2^{n-1}\) points and any embedded piecewise-linear triangulated disk bounded by \(K_n\) contains at least \(2^{n-1}\) triangles. In order to use results from smooth Morse theory and classification of diffeomorphisms of surfaces, the proof includes arguments for approximating piecewise-linear maps by smooth maps.

Reviewer: Mihai Cipu (Bucureşti)

MSC:

57M99	General low-dimensional topology
57M25	Knots and links in the \(3\)-sphere (MSC2010)
57Q99	PL-topology
57R12	Smooth approximations in differential topology
57R99	Differential topology

Keywords:

invariant train track; Morse theory; spanning disk