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29 March 2024
 
  » arxiv » math.GT/9801126

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A new algorithm for recognizing the unknot
Joan S. Birman ; Michael D. Hirsch ;
Date 28 Dec 1997
Journal Geom. Topol. 2 (1998), 175-220
Subject Geometric Topology MSC-class: 57M25, 57M50, 68Q15, 57M15, 68U05 | math.GT
AbstractThe 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.
Source arXiv, math.GT/9801126
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