Knot Theory with KnotPlot

Equilateral Stick Numbers

This research is in collaboration with Eric Rawdon of the Department of Mathematics at Duquesne University in Pittsburgh. We've used KnotPlot to find equilateral polygonal representatives of all knots to 10 crossings with the fewest number of sides. This number of sides is known as the equilateral stick number. It can be compared to the stick number, which is the same quantity when the constraint of being equilateral is dropped. Surprisingly, for 242 of the 249 prime knots examined, all have an equilateral stick number equal to their stick numbers.
Title: Upper Bounds for Equilateral Stick Numbers
Authors: Eric J. Rawdon and Robert G. Scharein
Abstract: We use algorithms in the software KnotPlot to compute upper bounds for the equilateral stick numbers of all prime knots through 10 crossings, i.e. the least number of equal length line segments it takes to construct a conformation of each knot type. We find seven knots for which we cannot construct an equilateral conformation with the same number of edges as a minimal non-equilateral conformation, notably the 819 knot.
Preprint: PDF (418 kBytes),Gzipped PostScript (293 kBytes) 
Appeared in: Contemporary Mathematics, Volume 304, 2002, pp55-75
Data: Gzipped TAR file of the data as it appears in the paper (integer coordinates).
Double precision coordinates are available as part of the KnotPlot distribution (in the sub-directories special/ms (non-equilateral) and special/mseq (equilateral) under the KnotPlot home. If you're just interested in the stick numbers themselves, consult the table of stick numbers of minimal stick knots.
Individual knots may be browsed online using the Knot Server or from the table of stick numbers.

3D view: Try clicking on one of the links in the table below to examine the exceptional knots in a 3D model viewer (requires Java). Use the links in the second and third columns to examine one knot at a time. Use the last column to view the two versions of the knot side by side in the same window.

Knot Equilateral
Both versions
819 9-stick eq 8-stick noneq both
929 10-stick eq 9-stick noneq both
1016 12-stick eq 11-stick noneq both
1079 12-stick eq 11-stick noneq both
10107 11-stick eq 10-stick noneq both
10119 11-stick eq 10-stick noneq both
10147 11-stick eq 10-stick noneq both

All the rest of the knots we found have a stick number equal to the equilateral stick number. Interactive 3D models for all of these can be accessed via the Knot Server link above, and for a select few from the table below.

Knot Equilateral
Both versions
31 6-stick eq 6-stick noneq both
41 7-stick eq 7-stick noneq both
10124 10-stick eq 10-stick noneq both
Square Knot 8-stick eq 8-stick noneq both
Granny Knot 8-stick eq 8-stick noneq both

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