Celebratio Mathematica — Birman

Interview with Joan Birman

Works connected to Michael David Hirsch

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J. S. Birman and M. D. Hirsch: “A new algorithm for recognizing the unknot,” Geom. Topol. 2 (1998), pp. 175–220. MR 1658024 Zbl 0955.57005 article

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.