Deformation of cylinder knots
Christoph Lamm
TL;DR
This paper investigates properties of cylinder knots parametrized by trigonometric functions, focusing on symmetry conditions and determinant equalities among Lissajous and billiard knots in a cylinder.
Contribution
It introduces the concept of billiard knots in a flat solid torus and explores their symmetry properties and determinant relationships, advancing understanding of these knot classes.
Findings
Conditions under which Z(s,n,m) equals Z(s,m,n)
Determinant equalities among certain Lissajous and billiard knots
Insights into symmetry properties of cylinder knots
Abstract
Knots parametrized in cylinder coordinates by t -> (st, 3 + cos(nt), cos(mt + \phi)) share properties of Lissajous and billiard knots in a cylinder. We use these 'billiard knots in a flat solid torus' to study two topics: when is Z(s,n,m) equal to Z(s,m,n)? And: why are the determinants of certain Lissajous and billiard knots in a cylinder equal?
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · semigroups and automata theory