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Pré-Publication, Document De Travail Année : 2016

Lissajous-toric knots

Marc Soret

Résumé

A point in the $(N,q)$-torus knot in $R^3$ goes q times along a vertical circle while this circle rotates $N$ times around the vertical axis. In the Lissajous-toric knot $K(N,q,p)$, the point goes along a vertical Lissajous curve (parametrized by $t↦(sin(qt+ϕ),cos(pt+ψ)))$ while this curve rotates $N$ times around the vertical axis. Such a knot has a natural braid representation $B_{N,q,p}$ which we investigate here. If $gcd(q,p)=1$, $K(N,q,p)$ is ribbon; if $gcd(q,p)=d>1$, $B_{N,q,p}$ is the $d$-th power of a braid which closes in a ribbon knot. We give an upper bound for the $4$-genus of $K(N,q,p)$ in the spirit of the genus of torus knots; we also give examples of $K(N,q,p)$'s which are trivial knots.

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Dates et versions

hal-01389949 , version 1 (30-10-2016)

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Citer

Marc Soret. Lissajous-toric knots. 2016. ⟨hal-01389949⟩
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