Satellites and Lorenz knots

Abstract

We construct infinitely many families of Lorenz knots that are satellites but not cables, giving counterexamples to a conjecture attributed to Morton. We amend the conjecture to state that Lorenz knots that are satellite have companion a Lorenz knot, and pattern equivalent to a Lorenz knot. We show this amended conjecture holds very broadly: it is true for all Lorenz knots obtained by high Dehn filling on a parent link, and other examples.

arXiv comment24 pages, 9 figures. V2: Details added to argument in section 5, formulae for patterns corrected in section 6, and half-twisting results removed from this paper. V3: Improved exposition throughout