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1998 A new algorithm for recognizing the unknot
Joan S Birman, Michael D Hirsch
Geom. Topol. 2(1): 175-220 (1998). DOI: 10.2140/gt.1998.2.175


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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Joan S Birman. Michael D Hirsch. "A new algorithm for recognizing the unknot." Geom. Topol. 2 (1) 175 - 220, 1998.


Received: 3 July 1997; Revised: 9 January 1998; Accepted: 4 January 1999; Published: 1998
First available in Project Euclid: 21 December 2017

zbMATH: 0955.57005
MathSciNet: MR1658024
Digital Object Identifier: 10.2140/gt.1998.2.175

Primary: 57M25, 57M50, 68Q15
Secondary: 57M15, 68U05

Rights: Copyright © 1998 Mathematical Sciences Publishers


Vol.2 • No. 1 • 1998
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