  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.
