2007/07/28 by Adam Boocher, Boocher, Adam, Jay Daigle +5 · 3 citations
Mathematics · #57M25 #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #math.GT #msc:57M25
paper · pdf · doi:10.48550/arxiv.0707.4210
34 pages, 5 figures, 18 tables
arxiv created 2007/07/28 · openalex publication_date 2007/07/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A Lissajous knot is one that can be parameterized by a single cosine function in each coordinate. Lissajous knots are highly symmetric, and for this reason, not all knots are Lissajous. We prove several theorems which 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 each coordinate is now described by a finite 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 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.