Braid ordering and the geometry of closed braid

TL;DR

This paper explores how braid ordering relates to the geometry of closed braids, establishing conditions under which the geometry and classification of the braid's closure are directly linked, and constructing hyperbolic knots from pseudo-Anosov braids.

Contribution

It proves a new connection between braid orderings and the geometry of braid closures, and demonstrates a method to construct hyperbolic knots from pseudo-Anosov braids.

Findings

Essential surfaces with small genus are circular-foliated when the Dehornoy floor is high.

Braid classification and the geometry of their closures are in one-to-one correspondence for large Dehornoy floors.

Explicitly constructs infinitely many hyperbolic knots from pseudo-Anosov elements.

Abstract

The relationships between braid ordering and the geometry of its closure is studied. We prove that if an essential closed surface $F$ in the complements of closed braid has relatively small genus with respect to the Dehornoy floor of the braid, $F$ is circular-foliated in a sense of Birman-Menasco's Braid foliation theory. As an application of the result, we prove that if Dehornoy floor of braids are larger than three, Nielsen-Thurston classification of braids and the geometry of their closure's complements are in one-to-one correspondence. Using this result, we construct infinitely many hyperbolic knots explicitly from pseudo-Anosov element of mapping class groups.