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Published November 25, 2009 | Submitted
Journal Article Open

Sampling Lissajous and Fourier Knots

Abstract

A Lissajous knot is one that can be parameterized as K(t)= (cos(n_x t+φ_x), cos(n_y t+φ_y) ,cos(n_z t+φ_z)), where the frequencies n_x, n_y, and n_z are relatively prime integers and the phase shifts φ_x, φ_y, and φ_z are real numbers. Lissajous knots are highly symmetric, and for this reason, not all knots are Lissajous. We prove several theorems that allow us to place bounds on the number of Lissajous knot types with given frequencies and to efficiently sample all possible Lissajous knots with a given set of frequencies. In particular, we systematically tabulate all Lissajous knots with small frequencies and as a result substantially enlarge the tables of known Lissajous knots. A Fourier-(i, j, k) knot is similar to a Lissajous knot except that the x, y, and z coordinates are now each described by a sum of i, j, and k cosine functions, respectively. According to Lamm, every knot is a Fourier-(1,1,k) knot for some k. By randomly searching the set of Fourier-(1,1,2) knots we find that all 2-bridge knots with up to 14 crossings are either Lissajous or Fourier-(1,1,2) knots. We show that all twist knots are Fourier-(1,1,2) knots and give evidence suggesting that all torus knots are Fourier-(1,1,2) knots. As a result of our computer search, several knots with relatively small crossing numbers are identified as potential counterexamples to interesting conjectures.

Additional Information

© 2009 A K Peters, Ltd. Received October 5, 2007; accepted in revised form August 20, 2008. First available in Project Euclid: 25 November 2009. This research was carried out at the Claremont College's REU program in the summer of 2006. The authors thank the National Science Foundation and the Claremont Colleges for their generous support.

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