Genus and braid index associated to sequences of renormalizable Lorenz maps

TL;DR

This paper explores the relationship between Lorenz links generated by renormalizable Lorenz maps and their topological invariants, providing explicit formulas for genus and braid index that grow exponentially with the map iterations.

Contribution

It introduces a new operation to generate knots and links from existing ones and derives explicit formulas for their genus and braid index in the context of renormalizable Lorenz maps.

Findings

Explicit formulas for genus and braid index of Lorenz knots

Growth of invariants is exponential with map iterations

Topological entropy remains constant on renormalization archipelagoes

Abstract

We describe the Lorenz links generated by renormalizable Lorenz maps with reducible kneading invariant $(K_f^-,K_f^+)=(X,Y)*(S,W)$, in terms of the links corresponding to each factor. This gives one new kind of operation that permits us to generate new knots and links from old. Using this result we obtain explicit formulas for the genus and the braid index of this renormalizable Lorenz knots and links. Then we obtain explicit formulas for sequences of these invariants, associated to sequences of renormalizable Lorenz maps with kneading invariant $(X,Y)*(S,W)^{*n}$, concluding that both grow exponentially. This is specially relevant, since it is known that topological entropy is constant on the archipelagoes of renormalization.