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 version	Non-equilateral version	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 version	Non-equilateral version	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

CoLab

My home page

KnotPlot Research Site

Knot theory