Allowable Sequences Arising from Double Polygons
by
Mark Fitch
Clemson University
Coauthors: Robert Jamison (Clemson University)

Allowable sequences are a tool developed by Goodman and Pollack that are used for studying convexity and slope problems in the plane. Note that an allowable sequence defines a rank 3 oriented matroid. Allowable sequences are sequences of special permutations. Those having a minimum number of permutations, called critical allowable sequences, are of special interest when studying slope problems.

Double polygons are point arrangements generalizing the pattern of the pentagram and Grunbaum double octagon. The allowable sequences of many double polygons can be tweaked to make them critical; however, the resulting allowable sequence no longer realizes a point arragement. The allowable sequences for other double polygons cannot be forced to be critical. This demonstrates restrictions on critical allowable sequences in general.

Date received: April 19, 2001