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Lissajous knots. (English) Zbl 0834.57005

A Lissajous knot is defined to be one isotopic to a knot which admits a parametrization \((x,y,z) = (\cos (n_xt + \varphi_x)\), \(\cos (n_yt + \varphi_y)\), \(\cos (n_zt + \varphi_z))\), where \(0 \leq t \leq 2 \pi\) and \(n_x, n_y, n_z\) are relatively prime integers. Motivation for considering Lissajous knots comes from the study of DNA molecular configurations and from Vassiliev theory. The authors made computer experiments, which suggest that Lissajous knots are quite rare, and they give the following criteria for a knot to be of Lissajous form: (1) If \(n_x, n_y, n_z\) are odd, then the Lissajous knot is strongly positive amphicheiral. (2) If one of \(n_x, n_y, n_z\) is even, then the Lissajous knot has period 2. (3) The Arf invariant of a Lissajous knot is 0.
Reviewer: M.Sakuma (Osaka)

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
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