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THE ALMOST ALTERNATING DIAGRAMS OF THE TRIVIAL KNOT
, 2006
"... Bankwitz characterized an alternating diagram representing the trivial knot. A nonalternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize an almost alternaing diagram representing the trivial knot. As a corollary we determine an unknot ..."
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Bankwitz characterized an alternating diagram representing the trivial knot. A nonalternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize an almost alternaing diagram representing the trivial knot. As a corollary we determine an unknotting number one alternating knot with a property that the unknotting operation can be done on its alternating diagram.
Towards an implementation of the BH algorithm for recognizing the unknot
 In KNOTS2000
, 2001
"... In the manuscript [2] the rst author and Michael Hirsch presented a thennew algorithm for recognizing the unknot. The rst part of the algorithm required the systematic enumeration of all discs which support a `braid foliation' and are embeddable in 3space. The boundaries of these `foliated ..."
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In the manuscript [2] the rst author and Michael Hirsch presented a thennew algorithm for recognizing the unknot. The rst part of the algorithm required the systematic enumeration of all discs which support a `braid foliation' and are embeddable in 3space. The boundaries of these `foliated embeddable discs' (FED's) are the collection of all closed braid representatives of the unknot, up to conjugacy, and the second part of the algorithm produces a word in the generators of the braid group which represents the boundary of the previously listed FED's. The third part tests whether a given closed braid is conjugate to the boundary of a FED on the list. In this paper we describe implementations of the rst and second parts of the algorithm. We also give some of the data which we obtained. The data suggests that FED's have unexplored and interesting structure. Open questions are interspersed throughout the manuscript. The third part of the algorithm was studied in [3] and [4], and implemented by S.J. Lee [20]. At this writing his algorithm is polynomial for n 4 and exponential for n 5. 1