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A PARTIAL ORDERING OF KNOTS AND LINKS THROUGH DIAGRAMMATIC UNKNOTTING

YUANAN DIAO

Department of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA

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CLAUS ERNST

Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA

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, and

ANDRZEJ STASIAK

Centre Intégratif de Génomique, Bâtiment Biophore, Niveau 1, Faculté de Biologie et de Médecine, Université de Lausanne, CH-1015 Lausanne-Dorigny, Switzerland

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https://doi.org/10.1142/S0218216509007026Cited by:6 (Source: Crossref)

Abstract

In this paper we define a partial ordering of knots and links using a special property derived from their minimal diagrams. A link is called a predecessor of a link if and a diagram of can be obtained from a minimal diagram D of by a single crossing change. In such a case, we say that . We investigate the sets of links that can be obtained by single crossing changes over all minimal diagrams of a given link. We show that these sets are specific for different links and permit partial ordering of all links. Some interesting results are presented and many questions are raised.

Keywords:

AMSC: Primary 57M25

References

C. C. Adams , The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots ( W. H. Freeman and Co. , 1994 ) . Google Scholar
A. Flammini and A. Stasiak, Proc. Royal Soc. A 463, 569 (2007). Crossref, Web of Science, Google Scholar
J. A. Bernhard, J. Knot Theory Ramifications 3(1), 1 (1994), DOI: 10.1142/S0218216594000022. Link, Google Scholar
S. A. Bleiler, Math. Proc. Cambridge Phil. Soc. 96, 469 (1984), DOI: 10.1017/S0305004100062381. Crossref, Web of Science, Google Scholar
G. Burde, Abh. Math. Sem. Univ. Hamburg 54, 199 (1984), DOI: 10.1007/BF02941452. Crossref, Web of Science, Google Scholar
G. Burde and H. Zieschang , Knots ( de Gruyter , Berlin , 1986 ) . Google Scholar
I. Darcy and D. M. W. Sumners, A strand passage metric for topoisomerase action, Knots '96: Proceedings of the Fifth MSJ International Research Institute of Mathematical Society of Japan, ed. S. Suzuki (World Scientific, 1997) pp. 267–278. Google Scholar
C. H. Dowker and M. Thistlethwaite, Topol. Appl. 16, 19 (1983), DOI: 10.1016/0166-8641(83)90004-4. Crossref, Web of Science, Google Scholar
J. Hoste and M. Thistlethwaite, Knotscape, http://www.math.utk.edu/~morwen/knotscape . Google Scholar
J. Hoste, M. Thistlethwaite and J. Weeks, Math. Intell. 20, 33 (1998), DOI: 10.1007/BF03025227. Crossref, Web of Science, Google Scholar
S. Jablin and R. Sazdanović, J. Knot Theory Ramifications 16(10), 1331 (2007). Link, Web of Science, Google Scholar
T. Kanenobu and H. Murakami, Proc. Amer. Math. Soc. 98(3), 499 (1986), DOI: 10.2307/2046210. Crossref, Web of Science, Google Scholar
P. Kohn, Proc. Amer. Math. Soc. 113, 1135 (1991), DOI: 10.2307/2048793. Crossref, Web of Science, Google Scholar
P. Kohn, Osaka J. Math. 30, 741 (1993). Web of Science, Google Scholar
P. B. Kronheimer and T. S. Mrowka, Topology 32(4), 773 (1993), DOI: 10.1016/0040-9383(93)90051-V. Crossref, Web of Science, Google Scholar
W. B. R. Lickorish and M. B. Thistlethwaite, Comment. Math. Helv. 63(4), 527 (1988), DOI: 10.1007/BF02566777. Crossref, Web of Science, Google Scholar
W. Menasco and M. Thistlethwaite, Bull. Amer. Math. Soc. 25, 403 (1991), DOI: 10.1090/S0273-0979-1991-16083-0. Crossref, Web of Science, Google Scholar
W. Menasco and M. Thistlethwaite, Ann. Math. 138, 113 (1993), DOI: 10.2307/2946636. Crossref, Web of Science, Google Scholar
K. Murasugi , Knot Theory and Its Applications ( Birkhäuser , Boston , 1996 ) . Google Scholar
Y. Nakanishi, Math. Seminar Notes Kobe Univ. 11(2), 257 (1983). Google Scholar
Y. Nakanishi, Rev. Mat. Univ. Complut. Madrid 9(2), 359 (1996). Google Scholar
S. Rankin et al., Knotilus, http://srankin.math.uwo.ca . Google Scholar
D. Rolfsen , Knots and Links , 2nd edn. , Mathematics Lecture Series 7 ( Publish or Perish, Inc. , Houston , 1990 ) . Google Scholar
L. Rudolph, Bull. Amer. Math. Soc. 29(1), 51 (1993), DOI: 10.1090/S0273-0979-1993-00397-5. Crossref, Web of Science, Google Scholar
A. Stoimenow, Math. Comput. 72(244), 2043 (2003). Crossref, Web of Science, Google Scholar
A. Stoimenow, Math. Scand. 94, 227 (2004). Crossref, Web of Science, Google Scholar
M. B. Thistlethwaite, Prime unoriented alternating links to 19 crossings, http://www.math.utk.edu/~morwen/png/link_stats.png . Google Scholar
I. Torisu, Proc. Amer. Math. Soc. 126(5), 1565 (1998), DOI: 10.1090/S0002-9939-98-04181-1. Crossref, Web of Science, Google Scholar
P. Traczyk, Contemporary Mathematics 233 (1999) pp. 215–220. Google Scholar