Volume 2, issue 1 (1998)

Download this article
For printing
Recent Issues

Volume 21
Issue 2, 647–1283
Issue 1, 1–645

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
A new algorithm for recognizing the unknot

Joan S Birman and Michael D Hirsch

Geometry & Topology 2 (1998) 175–220

arXiv: math.GT/9801126


The topological underpinnings are presented for a new algorithm which answers the question: “Is a given knot the unknot?” The algorithm uses the braid foliation technology of Bennequin and of Birman and Menasco. The approach is to consider the knot as a closed braid, and to use the fact that a knot is unknotted if and only if it is the boundary of a disc with a combinatorial foliation. The main problems which are solved in this paper are: how to systematically enumerate combinatorial braid foliations of a disc; how to verify whether a combinatorial foliation can be realized by an embedded disc; how to find a word in the the braid group whose conjugacy class represents the boundary of the embedded disc; how to check whether the given knot is isotopic to one of the enumerated examples; and finally, how to know when we can stop checking and be sure that our example is not the unknot.

knot, unknot, braid, foliation, algorithm
Mathematical Subject Classification
Primary: 57M25, 57M50, 68Q15
Secondary: 57M15, 68U05
Forward citations
Received: 3 July 1997
Revised: 9 January 1998
Accepted: 4 January 1999
Published: 4 January 1999
Proposed: David Gabai
Seconded: Wolfgang Metzler, Cameron Gordon
Joan S Birman
Mathematics Department
Columbia University
New York
New York 10027
Michael D Hirsch
Department of Computer Science
Emory University
Georgia 30322