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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