Geometry & Topology, Vol. 2 (1998) Paper no. 9, pages 175--220.

A new algorithm for recognizing the unknot

Joan S Birman, Michael D Hirsch

Abstract. 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.

Keywords. Knot, unknot, braid, foliation, algorithm

AMS subject classification. Primary: 57M25, 57M50, 68Q15. Secondary: 57M15, 68U05.

DOI: 10.2140/gt.1998.2.175

E-print: arXiv:math.GT/9801126

Submitted to GT on 3 July 1997. (Revised 9 January 1998.) Paper accepted 4 January 1999. Paper published 4 January 1999.

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Joan S Birman, Michael D Hirsch
Math Dept, Columbia University, NY, NY 10027, USA
Math and CS, Emory University, Atlanta, GA 30322, USA

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