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Unknotting and maximum unknotting numbers. (English) Zbl 1351.57015

Exploring new ways to unknot knots (which they name after themselves), these authors make remarkable advances in an area long studied with less than fully satisfying results. Their imaginary manipulations of actual knots and links yield a lengthy table of new unknotting numbers [Appendix B]. They also provide an excellent explanation of Conway’s notation, a brief discussion of Caudron’s 1982 thesis (the 1989 version of which is now available on N. J. A. Sloane’s OEIS website) and fresh depictions of the multiple writhes of the Perko pair knot.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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[1] 1. T. Abe and T. Kanenobu, Unoriented band surgery on knots and links, Kobe J. Math.31(1-2) (2014) 21-44, arXiv:1112.2449v1 [arXiv] . · Zbl 1358.57010
[2] 2. C. C. Adams, The Knot Book (Freeman, New York, 1994). · Zbl 0840.57001
[3] 3. Y. Bao, H(2)-unknotting operation related to 2-bridge links, Topology Appl.159(8) (2012) 2158-2167, arXiv:1104.4435v1 [arXiv] [math.GT]. genRefLink(16, ’S0218216516410108BIB003’, ’10.1016
[4] 4. D. Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Algebr. Geom. Topol.2 (2002) 337-370, arXiv:math/0201043v3 [arXiv] . genRefLink(16, ’S0218216516410108BIB004’, ’10.2140 · Zbl 0998.57016
[5] 5. J. A. Bernhard, Unknotting numbers and their minimal knot diagrams, J. Knot Theory Ramifications3(1) (1994) 1-5. [Abstract] · Zbl 0798.57009
[6] 6. S. A. Bleiler, A note on unknotting number, Math. Proc. Cambridge Phil. Soc.96 (1984) 469-471. genRefLink(16, ’S0218216516410108BIB006’, ’10.1017
[7] 7. A. Caudron, Classification des Noeuds et des Enlancements, Publications Mathematiques d’Orsay, Vol. 82 (Universite Paris-Sud Departement de Mathematique, Orsay, 1982). · Zbl 0505.57002
[8] 8. J. Conway, An enumeration of knots and links and some of their related properties, in Computational Problems in Abstract Algebra, ed. J. Leech, Proc. Conf. Oxford, 1967 (Pergamon Press, New York, 329-358, 1970). genRefLink(16, ’S0218216516410108BIB008’, ’10.1016
[9] 9. C. H. Dowker and M. B. Thistlethwaite, Classification of knot projections, Topology Appl.16 (1983) 19-31. genRefLink(16, ’S0218216516410108BIB009’, ’10.1016
[10] 10. T. Endo, T. Itoh and K. Taniyama, A graph-theoretic approach to a partial order of knots and links, Topology Appl.157(6) (2010) 1002-1010, arXiv:0806.3595v1 [arXiv] [mathGT]. genRefLink(16, ’S0218216516410108BIB010’, ’10.1016 · Zbl 1196.57004
[11] 11. J. Hoste and Y. Nakanishi and K. Taniyama, Unknotting operations involving trivial tangles, Osaka J. Math.27 (1990) 555-566. genRefLink(128, ’S0218216516410108BIB011’, ’A1990EF83100005’); · Zbl 0713.57006
[12] 12. J. Hoste and M. Thistlethwaite, Knotscape 1.01 (1999), http://www.math.utk.edu/orwen/knotscape.html.
[13] 13. S. Jablan, Unknotting and ascending numbers of knots and their families, arXiv:1107.2110v1 [math.GT].
[14] 14. S. Jablan and R. Sazdanović, LinKnot: Knot Theory by Computer, World Scientific edition ’Knots and Everything’, Vol. 21 (2007), pp. 500.
[15] 15. T. Kanenobu, H(2)-Gordian distance of knots, J. Knot Theory Ramifications20(6) (2011) 813-835. [Abstract] genRefLink(128, ’S0218216516410108BIB015’, ’000291945100003’); · Zbl 1230.57007
[16] 16. T. Kanenobu and H. Murakami, Two-bridge knots with unknotting number one, Proc. Amer. Math. Soc.98(3) (1986) 499-502. genRefLink(16, ’S0218216516410108BIB016’, ’10.1090
[17] 17. L. H. Kauffman and S. Lambropoulou, On the classification of rational knots, Enseign. Math.49 (2003) 357-410, arXiv:math.GT/0212011v2 [arXiv] . · Zbl 1059.57004
[18] 18. L. H. Kauffman and S. Lambropoulou, On the classification of rational tangles, Adv. Appl. Math.33(2) (2004) 199-237, arXiv:math.GT/0311499v2 [arXiv] . genRefLink(16, ’S0218216516410108BIB018’, ’10.1016 · Zbl 1057.57006
[19] 19. A. Kawauchi, A Survey of Knot Theory (Birkhuser, Basel, 1996). · Zbl 0861.57001
[20] 20. The Knot Atlas, DT (Dowker-Thistlethwaite) Codes, http://katlas.org/wiki/DT_(Dowker-Thistlethwaite)_Codes, retrieved October 2014.
[21] 21. P. Kohn, Two bridge links with unlinking number one, Proc. Amer. Math. Soc.98(4) (1991) 1135-1147. genRefLink(16, ’S0218216516410108BIB021’, ’10.1090 · Zbl 0734.57007
[22] 22. P. Kohn, Unlinking two component links, Osaka J. Math.30 (1993) 741-752. genRefLink(128, ’S0218216516410108BIB022’, ’A1993NC41600005’); · Zbl 0822.57003
[23] 23. J. C. Cha C. Livingston, Knot Info: Table of Knot Invariants, http://www.indiana. edu/notinfo/, accessed on October 2014.
[24] 24. Y. Miyazawa, Gordian distances and polynomial invariants, J. Knot Theory Ramifications20 (2011) 895-907. [Abstract] genRefLink(128, ’S0218216516410108BIB024’, ’000291945100007’); · Zbl 1221.57008
[25] 25. K. Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc.117 (1965) 387-422. genRefLink(16, ’S0218216516410108BIB025’, ’10.1090 · Zbl 0137.17903
[26] 26. Y. Nakanishi, A note on unknotting number, Math. Sem. Notes Kobe Univ.9 (1981) 99-108. · Zbl 0481.57002
[27] 27. Y. Nakanishi, Unknotting numbers and knot diagrams with the minimum crossings, Math. Sem. Notes Kobe Univ.11 (1983) 257-258. · Zbl 0549.57003
[28] 28. J. Przytycki, From Goeritz matrices to quasi-alternating links, Invent. Math.182(2) (2010) 419-447, [http://arxiv.org/abs/0909.1118] . genRefLink(16, ’S0218216516410108BIB028’, ’10.1007
[29] 29. J. Rasmussen, Khovanov homology and the slice genus, Inventiones Mathematicae182 (2010) 479-497. genRefLink(16, ’S0218216516410108BIB029’, ’10.1007
[30] 30. D. Rolfsen, Knots and Links (Publish & Perish Inc., Berkeley, 1976); American Mathematical Society (AMS Chelsea Publishing, 2003). genRefLink(16, ’S0218216516410108BIB030’, ’10.1090
[31] 31. H. Schubert, Knoten mit zwei Brücken, Math. Z.65 (1956) 133-170. genRefLink(16, ’S0218216516410108BIB031’, ’10.1007 · Zbl 0071.39002
[32] 32. A. Stoimenow, On the unknotting number of minimal diagrams, Math. Comput.72(244) (2003) 2043-2057. genRefLink(16, ’S0218216516410108BIB032’, ’10.1090 · Zbl 1025.57014
[33] 33. A. Stoimenow, On unknotting numbers and knot trivadjacency, Math. Scand.94 (2004) 227-248. · Zbl 1060.57008
[34] 34. K. Taniyama, A partial order of knots, Tokyo J. Math.12 (1989) 205-229. genRefLink(16, ’S0218216516410108BIB034’, ’10.3836 · Zbl 0688.57006
[35] 35. K. Taniyama, A partial order of links, Tokyo J. Math.12 (1989) 475-484. genRefLink(16, ’S0218216516410108BIB035’, ’10.3836 · Zbl 0698.57001
[36] 36. P. Traczyk, A combinatorial formula for the signature of alternating diagrams, Fund. Math.184 (2004) 311-316 (a new version of the unpublished manuscript, 1987). · Zbl 1064.57009
[37] 37. A. Zekovic, Conway notation and its appliance in knot distance determination methods in knot theory, PhD thesis, University of Belgrade (2015).
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