No Access

UNKNOTTING NUMBERS AND TRIPLE POINT CANCELLING NUMBERS OF TORUS-COVERING KNOTS

INASA NAKAMURA

Department of Mathematics, Gakushuin University, 1-5-1 Mejiro Toshima-ku, Tokyo 171-8588, Japan

https://doi.org/10.1142/S0218216513500107Cited by:0 (Source: Crossref)

Abstract

It is known that any surface knot can be transformed to an unknotted surface knot or a surface knot which has a diagram with no triple points by a finite number of 1-handle additions. The minimum number of such 1-handles is called the unknotting number or the triple point cancelling number, respectively. In this paper, we give upper bounds and lower bounds of unknotting numbers and triple point cancelling numbers of torus-covering knots, which are surface knots in the form of coverings over the standard torus T. Upper bounds are given by using m-charts on T presenting torus-covering knots, and lower bounds are given by using quandle colorings and quandle cocycle invariants.

Keywords:

AMSC: 57Q45, 57Q35

References

S. Asami and S. Satoh, Bull. London Math. Soc. 37, 285 (2005), DOI: 10.1112/S0024609304003832. Crossref, Web of Science, Google Scholar
J. Boyle, Trans. Amer. Math. Soc. 306, 475 (1988). Crossref, Web of Science, Google Scholar
E. Brieskorn, Automorphic sets and singularities, Braids: Proc. Summer Research Conf.78, Contemporary Mathematics (American Mathematical Society Providence, RI, 1988) pp. 45–115. Google Scholar
J. S. Carteret al., Trans. Amer. Math. Soc. 355, 3947 (2003), DOI: 10.1090/S0002-9947-03-03046-0. Crossref, Web of Science, Google Scholar
J. S. Carter, S. Kamada and M. Saito, J. Knot Theory Ramifications 10, 345 (2001), DOI: 10.1142/S0218216501000901. Link, Web of Science, Google Scholar
J. S. Carter , S. Kamada and M. Saito , Surfaces in 4- space , Encyclopaedia of Mathematical Sciences 142 ( Springer-Verlag , Berlin , 2004 ) . Crossref, Google Scholar
J. S. Carter and M. Saito , Knotted Surfaces and Their Diagrams , Mathematical Surveys and Monographs 55 ( American Mathematical Society Providence , RI , 1998 ) . Google Scholar
R. H. Fox, Topology of 3- Manifolds, ed. M. K. Fort Jr. (Prentice-Hall, Englewood Cliffs, NJ, 1962) pp. 120–167. Google Scholar
C. F. Gauss, Disquisitiones Arithmeticae, translated by A. A. Clarke, revised by W. C. Waterhouse with the help of C. Greither and A. W. Grootendorst (Springer-Verlag, NY, 1986) . Google Scholar
F. Hosokawa and A. Kawauchi, Osaka J. Math. 16, 233 (1979). Web of Science, Google Scholar
F. Hosokawa, T. Maeda and S. Suzuki, Math. Sem. Notes Kobe Univ. 7, 409 (1979). Google Scholar
A. Inoue and Y. Kabaya, Quandle homology and complex volume, preprint (2010) , arXiv:1012.2923v2 . Google Scholar
K. Ireland and M. Rosen , A Classical Introduction to Modern Number Theory , Graduate Texts in Mathematics 84 ( Springer-Verlag , NY , 1990 ) . Crossref, Google Scholar
M. Iwakiri, Topology Appl. 153, 2815 (2006), DOI: 10.1016/j.topol.2005.12.001. Crossref, Web of Science, Google Scholar
D. Joyce, J. Pure Appl. Algebra 23, 37 (1982), DOI: 10.1016/0022-4049(82)90077-9. Crossref, Web of Science, Google Scholar
S. Kamada, J. Knot Theory Ramifications 1, 137 (1992), DOI: 10.1142/S0218216592000082. Link, Google Scholar
S. Kamada, 2-dimensional braids and chart descriptions, in Topics in Knot Theory NATO Advanced Science Institute Series C: Mathematical and Physical Sciences, Vol. 399 (Kluwer Academic Publishers Dordrecht, 1993), pp. 277–287 . Google Scholar
S. Kamada, Topology 33, 113 (1994), DOI: 10.1016/0040-9383(94)90038-8. Crossref, Web of Science, Google Scholar
S. Kamada, J. Knot Theory Ramifications 4, 517 (1996). Google Scholar
S. Kamada, Osaka J. Math. 36, 33 (1999). Web of Science, Google Scholar
S. Kamada , Braid and Knot Theory in Dimension Four , Mathematical Surveys and Monographs 95 ( American Mathematical Society , Providence, RI , 2002 ) . Crossref, Google Scholar
T. Kanenobu, J. Knot Theory Ramifications 5, 171 (1996), DOI: 10.1142/S0218216596000126. Google Scholar
T. Kanenobu and Y. Marumoto, Osaka J. Math. 34, 525 (1997). Web of Science, Google Scholar
A. Kawauchi, J. Knot Theory Ramifications 11, 1043 (2002), DOI: 10.1142/S0218216502002128. Link, Web of Science, Google Scholar
S. Matveev, Math. USSR-Sb. 47, 73 (1982), DOI: 10.1070/SM1984v047n01ABEH002630. Crossref, Web of Science, Google Scholar
K. Miyazaki, Kobe J. Math. 3, 77 (1986). Google Scholar
T. Mochizuki, J. Pure Appl. Algebra 179, 287 (2003), DOI: 10.1016/S0022-4049(02)00323-7. Crossref, Web of Science, Google Scholar
I. Nakamura, Algebr. Geom. Topol. 11, 1497 (2011), DOI: 10.2140/agt.2011.11.1497. Crossref, Web of Science, Google Scholar
T. Nosaka, Topol. Appl. 158(8), 996 (2011), DOI: 10.1016/j.topol.2011.02.006. Crossref, Web of Science, Google Scholar
L. Rudolph, Comment. Math. Helv. 58(1), 1 (1983), DOI: 10.1007/BF02564622. Crossref, Web of Science, Google Scholar
S. Satoh, Kobe J. Math. 21(2), 71 (2004). Web of Science, Google Scholar
M. Takasaki, Tohoku Math. J. 49, 145 (1943). Web of Science, Google Scholar