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A new algorithm for recognizing the unknot
Joan S Birman and Michael D Hirsch
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Geometry & Topology 2 (1998)
175–220
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Abstract
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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.
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Keywords
knot, unknot, braid, foliation,
algorithm
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Mathematical Subject Classification
Primary: 57M25, 57M50, 68Q15
Secondary: 57M15, 68U05
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Publication
Received: 3 July 1997
Revised: 9 January 1998
Accepted: 4 January 1999
Published: 4 January 1999
Proposed: David Gabai
Seconded: Wolfgang Metzler, Cameron Gordon
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