1998/01/31 by Joan S. Birman, Michael D. Hirsch · 2 citations
Mathematics · #math.GT #msc:57M25 #msc:57M50 #msc:68Q15 #msc:57M15 #msc:68U05
paper · pdf · doi:10.2140/gt.1998.2.175
published as Geom. Topol. 2 (1998) 175-220 · 46 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol2/paper9.abs.html
arxiv created 1999/01/04 · arxiv updated 2014/11/11
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.