THE GENUS OF CLOSED 3-BRAIDS

The problem of finding the genus of a given link type of braid index 3 is solved by constructive methods. The results depend upon a new solution to the conjugacy problem in B₃. In this solution, conjugacy classes are represented by shortest words in terms of cyclically symmetric elementary braids which are used as generators in the new presentation of B₃. By related results of Bennequin [2] and of Birman and Menasco [4], the minimal spanning surfaces are described by these shortest words. An effective algorithm is given to find these shortest words starting with an arbitrary 3-braid representative of the link. It is proved that up to an easily described family of exceptional cases, a link of braid index 3 has a unique isotopy class of minimal spanning surfaces. The growth functions of B₃ and its conjugacy classes are also given here.