No Access

UNKNOTTING NUMBERS OF DIAGRAMS OF A GIVEN NONTRIVIAL KNOT ARE UNBOUNDED

KOUKI TANIYAMA

Department of Mathematics, School of Education, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo, 169-8050, Japan

Search for more papers by this author

https://doi.org/10.1142/S0218216509007361Cited by:7 (Source: Crossref)

Abstract

We show that for any nontrivial knot K and any natural number n, there is a diagram D of K such that the unknotting number of D is greater than or equal to n. It is well-known that twice the unknotting number of K is less than or equal to the crossing number of K minus one. We show that the equality holds only when K is a (2, p)-torus knot.

Dedicated to Professor Akio Kawauchi for his 60th birthday

Keywords:

AMSC: 57M25

References

C. Adams , The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots ( W. H. Freeman and Co. , 1994 ) . Google Scholar
J. Bernhard, J. Knot Theory Ramifications 3, 1 (1994), DOI: 10.1142/S0218216594000022. Link, Google Scholar
S. Bleiler, Math. Proc. Cambridge Phil. Soc. 96, 469 (1984), DOI: 10.1017/S0305004100062381. Crossref, Web of Science, Google Scholar
J. Conway, Computational Problems in Abstract Algebra (Pergamon Press, Oxford, 1967) pp. 329–358. Google Scholar
Y. Diao, C. Ernst and A. Stasiak, A partial ordering of knots through diagrammatic unknotting, preprint . Google Scholar
S. Jablan, Filomat 12, 113 (1998). Google Scholar
S. Jablan and R. Sazdanovic, J. Knot Theory Ramifications 16, 1331 (2007), DOI: 10.1142/S0218216507005828. Link, Web of Science, Google Scholar
P. Kohn, Proc. Amer. Math. Soc. 113, 1135 (1991), DOI: 10.2307/2048793. Crossref, Web of Science, Google Scholar
Y. Nakanishi, Math. Sem. Notes Kobe Univ. 11, 257 (1983). Google Scholar
Y. Nakanishi, Proc. Amer. Math. Soc. 124, 1625 (1996), DOI: 10.1090/S0002-9939-96-03453-3. Crossref, Web of Science, Google Scholar
M. Ozawa, Ascending number of knots and links, preprint , arXiv:0705.3337 . Google Scholar
H. Schubert, Math. Z. 65, 133 (1956), DOI: 10.1007/BF01473875. Crossref, Google Scholar
A. Stoimenow, J. London Math. Soc. 63, 487 (2001), DOI: 10.1017/S0024610700001897. Crossref, Web of Science, Google Scholar
T. Tsukamoto, J. Topol. 2, 77 (2009), DOI: 10.1112/jtopol/jtp001. Crossref, Web of Science, Google Scholar