TOWARDS AN IMPLEMENTATION OF THE B–H ALGORITHM FOR RECOGNIZING THE UNKNOT

In the manuscript [2] the first author and Michael Hirsch presented a then-new algorithm for recognizing the unknot. The first part of the algorithm required the systematic enumeration of all discs which support a 'braid foliation' and are embeddable in 3-space. 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 first 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 question 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]  Bemerkungen zur knotentheorie , 1934 .

[2]  W. Haken.  Theorie der Normalflächen , 1961 .

[3]  W. Magnus,et al.  On a theorem of V. I. arnol'd , 1969 .

[4]  Donald B. Johnson,et al.  Finding All the Elementary Circuits of a Directed Graph , 1975, SIAM J. Comput..

[5]  G. Baumslag.  Free subgroups of certain one-relator groups defined by positive words , 1983, Mathematical Proceedings of the Cambridge Philosophical Society.

[6]  Hugh R. Morton,et al.  An irreducible 4-string braid with unknotted closure , 1983, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  H. Morton.  Threading knot diagrams , 1986, Mathematical Proceedings of the Cambridge Philosophical Society.

[8]  Shuji Yamada.  The minimal number of Seifert circles equals the braid index of a link , 1987 .

[9]  Pierre Vogel,et al.  Representation of links by braids: A new algorithm , 1990 .

[10]  T. Fiedler.  A SMALL STATE SUM FOR KNOTS , 1993 .

[11]  William Jaco,et al.  Algorithms for the complete decomposition of a closed $3$-manifold , 1995 .

[12]  Ki Hyoung Ko,et al.  Band-generator presentation for the 4-braid group , 1997 .

[13]  Joan S. Birman,et al.  A new approach to the word and conjugacy problems in the braid groups , 1997 .

[14]  K. Ko,et al.  Genera of some closed 4-braids , 1997 .

[15]  J. Lagarias,et al.  The number of Reidemeister moves needed for unknotting , 1998, math/9807012.

[16]  Joel Hass,et al.  Algorithms for recognizing knots and 3-manifolds , 1998 .

[17]  L. Rudolph.  Quasipositive plumbing (constructions of quasipositive knots and links, V) , 1998 .

[18]  J. Weeks,et al.  The first 1,701,936 knots , 1998 .

[19]  Michael D. Hirsch,et al.  A new algorithm for recognizing the unknot , 1998 .

[20]  STUDYING SURFACES VIA CLOSED BRAIDS , 1998, math/9804028.

[21]  Frank Ruskey,et al.  An efficient algorithm for generating necklaces with fixed density , 1999, SODA '99.

[22]  Jeffrey C. Lagarias,et al.  The computational complexity of knot and link problems , 1999, JACM.

[23]  A foliated disk whose boundary is Morton's irreducible 4-braid , 2000, Mathematical Proceedings of the Cambridge Philosophical Society.