#### 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.

Notes on file formats
Joan S Birman, Michael D Hirsch

Math Dept, Columbia University, NY, NY 10027, USA

Math and CS, Emory University, Atlanta, GA 30322, USA

Email: jb@math.columbia.edu, hirsch@mathcs.emory.edu

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