No Access

Covering diagrams over surface-knot diagrams

Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P. O. Box 36, PC123, Al-Khoudh, Muscat, Sultanate of Oman

E-mail Address: tyashiro@cobalt.net.tokai-u.jp

Search for more papers by this author

https://doi.org/10.1142/S0218216518500426Cited by:0 (Source: Crossref)

Abstract

A surface-knot is a closed oriented surface smoothly embedded in 4-space and a surface-knot diagram is a projected image of a surface-knot under the orthogonal projection in 3-space with crossing information. Every surface-knot diagram induces a rectangular-cell complex. In this paper, we introduce a covering diagram over a surface-knot diagram. the covering map induces a covering of the rectangular-cell complexes. As an application, a lower bound of triple point numbers for a family of surface-knots is obtained.

Keywords:

AMSC: 57Q45, 57M25

References

1. J. S. Carter and M. Saito, Knotted Surfaces and Their Diagrams, Mathematical Surveys and Monographs, Vol. 55 (American Mathematical Society, Providence, RI, 1998). Google Scholar
2. J. S. Carter and M. Saito, Surfaces in 3-space that do not lift to embeddings in 4-space, in Knot Theory, Banach Center Publications, Vol. 42, Institute of Mathematics Polish Academy of Sciences (Warsaw, 1998), pp. 29–47. Crossref, Google Scholar
3. A. Kawauchi, On pseudo-ribbon surface-links, J. Knot Theory Ramifications 11 (2002) 1043–1062. Link, Web of Science, Google Scholar
4. A. A. Kharusi and T. Yashiro, No orientable surface-knot with genus one has the triple point number two, preprint, arXiv:1506.01489v8 [math.AT]. Google Scholar
5. A. Mohamad and T. Yashiro, On construction of surface-knots, J. Knot Theory Ramifications 25 (2016) 1650053. Link, Web of Science, Google Scholar
6. D. Roseman, Reidmeister-type moves for surfaces in four dimensional space, in Knot Theory, Vol. 42 (Banach Center Publications, 1998), pp. 347–380. Crossref, Google Scholar
7. S. Satoh, No 2-knot has triple point number two or three, Osaka J. Math. 42 (2005) 543–556. Web of Science, Google Scholar
8. S. Satoh and A. Shima, The 2-twist-spun trefoil has the triple point number four, Trans. Amer. Math. Soc. 356 (2004) 1007–1024. Crossref, Web of Science, Google Scholar
9. S. Satoh and A. Shima, Triple point numbers and quandle cocycle invariants of knotted surfaces in 4-space, New Zealand J. Math. 34 (2005) 71–79. Google Scholar
10. T. Yashiro, A note on Roseman moves, Kobe J. Math. 22 (2005) 31–38. Google Scholar
11. T. Yashiro, Cell-complexes for $t$ -minimal surface diagrams with 4 triple points, Intelligence of Low Dimensional Topology 2006: Hiroshima, Japan, 22–26 July 2006 ( J. Scott Carter et al., ed.), K&E Series on Knots and Everything (World Scientific 2007), pp. 367–374. Google Scholar
12. T. Yashiro, Pseudo-cycles of surface-knots, J. Knot Theory Ramifications 25 (2016) 1650068. Link, Web of Science, Google Scholar
13. E. C. Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965) 471–495. Crossref, Web of Science, Google Scholar