Genus and braid index associated to sequences of renormalizable Lorenz maps

2008/03/27 by Nuno Franco, Luis Silva, Franco, Nuno +1
Mathematics · #37E05 #37E20 #57M25 #57M27 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #math.DS #math.GT #msc:37E05 #msc:37E20 #msc:57M25 #msc:57M27

paper · pdf · doi:10.48550/arxiv.0803.3895

arxiv created 2009/01/08 · arxiv updated 2009/12/01

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.

Related

• Farey neighbors and hyperbolic Lorenz knots
• Lorenz links, T-links, Minimal Braids, Positive braids with a full twist, and Geometric Types
• Knot Theory: from Fox 3-colorings of links to Yang-Baxter homology and Khovanov homology
• Quantum invariants of links and new quantum field models
• Renormalization for Lorenz maps of monotone combinatorial types