A new algorithm for unknots
by
Michael D. Hirsch
Emory University
Coauthors: Joan S. Birman

It is well understood that every knot or link in 3-space bounds an orientable surface. When the link in question is a closed braid, the surface inherits a "braid foliation" which is the pull back of the öpen book foliation of 3-space. These braid foliations were first studied by Bennequin and much further work was done by Birman and Menasco. Braid foliations are determined by their singular leaves and a small amount of combinatorial data; they are fundamentally a combinatorial object, and thus well suited for algorithmic uses, though this has not been done previously. In this work we wish to use such foliations to algorithmically study knots and especially unknots. To do so we need to understand three things about such foliations:First, how to represent such a foliation as a combinatorial object; Second, how to determine if a given foliation comes about as a braid foliation, i.e., given a foliated disc, is there an embedding of the disc in 3-space for which the foliation is the pullback of the open book foliation of 3-space; and third, given an braid foliation, what is the boundary word that is the boundary braid of the embedded disc realizing this foliation. The solutions to these problems turn out to be surprisingly simple and elegant. Solving these problems allows us to develop a new algorithm for solving the knottedness problem. Our algorithm uses braid foliations and is completely different from Haken's original algorithm.

Date received: February 12, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caas-64.