Geometry & Topology

A new algorithm for recognizing the unknot

Joan S Birman and Michael D Hirsch

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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.

Article information

Geom. Topol., Volume 2, Number 1 (1998), 175-220.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M50: Geometric structures on low-dimensional manifolds 68Q15: Complexity classes (hierarchies, relations among complexity classes, etc.) [See also 03D15, 68Q17, 68Q19]
Secondary: 57M15: Relations with graph theory [See also 05Cxx] 68U05: Computer graphics; computational geometry [See also 65D18]

knot unknot braid foliation algorithm


Birman, Joan S; Hirsch, Michael D. A new algorithm for recognizing the unknot. Geom. Topol. 2 (1998), no. 1, 175--220. doi:10.2140/gt.1998.2.175.

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