Braid Ordering and the Geometry of Closed Braid

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.