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MINIMAL EDGE PIECEWISE LINEAR KNOTS J. A. CALVO
 

Summary: MINIMAL EDGE PIECEWISE LINEAR KNOTS
J. A. CALVO
Department of Mathematics, University of California,
Santa Barbara, CA 93106, USA
K. C. MILLETT
Department of Mathematics, University of California,
Santa Barbara, CA 93106, USA
The space of n­sided polygons embedded in three­space consists of a smooth man­
ifold in which points correspond to piecewise linear or ``geometric'' knots, while
paths correspond to isotopies which preserve the geometric structure of these knots.
Two cases are considered: (i) the space of polygons with varying edge length, and
(ii) the space of equilateral polygons with unit­length edges. In each case, the
spaces are explored via a Monte Carlo search to estimate the distinct knot types
represented. Preliminary results of these searches are presented. Additionally, this
data is analyzed to determine the smallest number of edges necessary to realize
each knot type with nine or fewer crossings as a polygon, i.e. its ``minimal stick
number.''
1. Introduction, vocabulary, and history of geometric knots.
The topological and geometric knotting of circles occurs in many contexts in
the natural sciences. 17,25 By geometric knotting we mean the imposition of geo­

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics