TOWARDS AN IMPLEMENTATION OF THE B–H ALGORITHM FOR RECOGNIZING THE UNKNOT

Abstract

In the manuscript [2] the first author and Michael Hirsch presented a then-new algorithm for recognizing the unknot. The first part of the algorithm required the systematic enumeration of all discs which support a 'braid foliation' and are embeddable in 3-space. The boundaries of these 'foliated embeddable discs' (FED's) are the collection of all closed braid representatives of the unknot, up to conjugacy, and the second part of the algorithm produces a word in the generators of the braid group which represents the boundary of the previously listed FED's. The third part tests whether a given closed braid is conjugate to the boundary of a FED on the list. In this paper we describe implementations of the first and second parts of the algorithm. We also give some of the data which we obtained. The data suggests that FED's have unexplored and interesting structure. Open question are interspersed throughout the manuscript. The third part of the algorithm was studied in [3] and [4], and implemented by S. J. Lee [20]. At this writing his algorithm is polynomial for n ≤ 4 and exponential for n ≥ 5.