From: Steve Gray Subject: Knotted polygon Date: Mon, 06 Aug 2001 23:30:34 GMT Newsgroups: sci.math Summary: Algebraic characterization of piecewise-linear knots Question: How to characterize algebraically an ordered set of 6 points (XiYiZi), 0<=i<=5, in 3-D such that when connected with straight segments the resulting polygon is knotted (that is, cannot be continuously deformed into a convex plane polygon with no crossings). (Only "overhand" knots are possible with six segments.) I'm posting this to two other groups. -- Respect reality. It's all you've got. ============================================================================== [NB -- It is possible to compute the fundamental group of the knot, which will be nontrivial iff this polygon is knotted. The computation requires determining which of the 15 pairs of line segments have crossed projections in the xy plane, and for each such pair determining which passes "over" and which passes "under". From this data one may record the Wirtinger presentation of the fundamental group and then any derivative algebraic invariants (e.g. the knot polynomials). So one may decide algebraically whether the six segments are knotted. Whether this constitutes an adequate "characterization" is unclear. --djr]