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Birman, Joan S.; Rampichini, Marta; Boldi, Paolo; Vigna, Sebastiano

Towards an implementation of the B-H algorithm for recognizing the unknot. (English) Zbl 1007.57004

J. Knot Theory Ramifications 11, No. 4, 601-645 (2002).
This paper is concerned with implementation of an algorithm of Joan S. Birman and Michael D. Hirsch [Geom. Topol. 2, 175-220 (1998; Zbl 0955.57005)] to determine whether or not a knot is unkotted.
The foundation for the algorithm is foliation of an imbedded disc with boundary an unknot. The foliation arises from the intersection of the half planes with \(z\)-axis as boundary and a disc bounded by a closed braid around the \(z\)-axis.
With an ordering on the foliation types, it is possible to order the conjugacy classes of the \(n\) strand braid representatives of the unknot. Then if the knot to be tested does not lie among a finite set of conjugacy classes one may conclude that the knot is not the unknot.
Computer programs, numerical results, and extensive analyses of the singularities of the foliations are included, along with numerous questions.
Reviewer: Lee P.Neuwirth (Princeton)

Cited in 2 Documents

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
68W05 Nonnumerical algorithms
57M50 General geometric structures on low-dimensional manifolds
57-04 Software, source code, etc. for problems pertaining to manifolds and cell complexes

Keywords:

unknot; algorithm; braid; foliation

Citations:

Zbl 0955.57005
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Full Text: DOI arXiv

References:

[1] DOI: 10.2140/gt.1998.2.175 · Zbl 0955.57005 · doi:10.2140/gt.1998.2.175
[2] DOI: 10.1006/aima.1998.1761 · Zbl 0937.20016 · doi:10.1006/aima.1998.1761
[3] DOI: 10.2307/2153953 · Zbl 0758.57005 · doi:10.2307/2153953
[4] DOI: 10.1016/0040-9383(93)90020-V · Zbl 0787.57007 · doi:10.1016/0040-9383(93)90020-V
[5] DOI: 10.1007/BF02940674 · Zbl 0009.23001 · doi:10.1007/BF02940674
[6] DOI: 10.1007/BF02559591 · Zbl 0100.19402 · doi:10.1007/BF02559591
[7] DOI: 10.1016/S0960-0779(97)00109-4 · Zbl 0935.57014 · doi:10.1016/S0960-0779(97)00109-4
[8] DOI: 10.1090/S0894-0347-01-00358-7 · Zbl 0964.57005 · doi:10.1090/S0894-0347-01-00358-7
[9] DOI: 10.1145/301970.301971 · Zbl 1065.68667 · doi:10.1145/301970.301971
[10] DOI: 10.1007/BF03025227 · Zbl 0916.57008 · doi:10.1007/BF03025227
[11] Jaco W., III J. Math 39 pp 358– (1995)
[12] Johnson D. B., SI AM J. Cornput. 4 pp 77– (1975)
[13] DOI: 10.1016/S0166-8641(96)00148-4 · Zbl 0879.57005 · doi:10.1016/S0166-8641(96)00148-4
[14] Kneser H., Jahresbericht Math. Verein. 28 pp 248– (1929)
[15] DOI: 10.1016/S0166-8641(96)00149-6 · Zbl 0879.57006 · doi:10.1016/S0166-8641(96)00149-6
[16] DOI: 10.1002/cpa.3160220508 · Zbl 0184.49001 · doi:10.1002/cpa.3160220508
[17] DOI: 10.1017/S0305004100060527 · Zbl 0512.20015 · doi:10.1017/S0305004100060527
[18] DOI: 10.1017/S0305004100060540 · Zbl 0522.57006 · doi:10.1017/S0305004100060540
[19] DOI: 10.1090/S0002-9939-98-04407-4 · Zbl 0888.57010 · doi:10.1090/S0002-9939-98-04407-4
[20] DOI: 10.1137/S0097539798344112 · Zbl 0937.68091 · doi:10.1137/S0097539798344112
[21] DOI: 10.1007/BF02566597 · Zbl 0703.57004 · doi:10.1007/BF02566597
[22] DOI: 10.1007/BF01389082 · Zbl 0634.57004 · doi:10.1007/BF01389082
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