Studying links via closed braids I: A finiteness theorem

This paper is the first in a series which study the closed braid representatives of an oriented link type L in oriented 3-space. A combinatorial symbol is introduced which determines an oriented spanning surface F for a representative L of L. The surface F is in a special position in 3-space relative to the braid axis A and the fibers in a fibration of the complement of A. The symbol simultaneously describes F as an embedded surface and L as a closed braid. Therefore it is both geometrically and algebraically meaningful. Using it, a complexity function is introduced. It is proved that L is described by at most finitely many combinatorial symbols, and thus by finitely many conjugacy classes in each braid group B_n when the complexity is minimal.