Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T02:07:34.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Regular-equivalence of 2-knot diagrams and sphere eversions

Published online by Cambridge University Press:  05 May 2016

MASAMICHI TAKASE  and
KOKORO TANAKA
MASAMICHI TAKASE
Affiliation:
Faculty of Science and Technology, Seikei University, 3-3-1 Kichijoji-kitamachi, Musashino, Tokyo 180-8633, Japan. e-mails: [email protected]
KOKORO TANAKA
Affiliation:
Department of Mathematics, Tokyo Gakugei University, 4-1-1 Nukuikita-machi, Koganei, Tokyo 184-8501, Japan. e-mails: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

For each diagram D of a 2-knot, we provide a way to construct a new diagram D′ of the same knot such that any sequence of Roseman moves between D and D′ necessarily involves branch points. The proof is done by developing the observation that no sphere eversion can be lifted to an isotopy in 4-space.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 2 , September 2016 , pp. 237 - 246
Copyright
Copyright © Cambridge Philosophical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Aitchison, I. R. The Holiverse: holistic eversion of the 2-sphere, preprint 2010, available at arXiv:1008.0916.Google Scholar
[2] Carter, J. S. An excursion in diagrammatic algebra: turning a sphere from red to blue. Ser. Knots Everything. 48 (2012), (World Scientific Publishing).Google Scholar
[3] Carter, J. S. and Saito, M. Cancelling branch points on projections of surfaces in 4-space. Proc. Amer. Math. Soc. 116 (1992), no. 1, 229–237.Google Scholar
[4] Francis, G. K. and Morin, B. Arnold Shapiro's eversion of the sphere. Math. Intelligencer 2 (1979/80), 200–203.Google Scholar
[5] Freedman, M. H. Quadruple points of 3-manifolds in S 4 . Comment. Math. Helv. 53 (1978), 385–394.CrossRefGoogle Scholar
[6] Hirsch, M. W. Immersions of manifolds. Trans. Amer. Math. Soc. 93 (1959), 242–276.Google Scholar
[7] Hughes, J. F. and Melvin, P. M. The Smale invariant of a knot. Comment. Math. Helv. 60 (1985), 615–627.Google Scholar
[8] Hughes, J. F. Another proof that every eversion of the sphere has a quadruple point. Amer. J. Math. 107 (1985), 501–505.Google Scholar
[9] Hughes, J. F. Bordism and regular homotopy of low-dimensional immersions. Pacific J. Math. 156 (1992), 155–184.Google Scholar
[10] Jabłonowski, M. Knotted surfaces and equivalencies of their diagrams without triple points. J. Knot Theory Ramifications 21 (2012), 1250019 (6 pages).Google Scholar
[11] Kaiser, U. Immersions in codimension 1 up to regular homotopy. Arch. Math. (Basel) 51 (1988), 371–377.Google Scholar
[12] Kawamura, K. On relationship between seven types of Roseman moves. Topology Appl. 196 (2015), part B, 551–557.Google Scholar
[13] Kawamura, K., Oshiro, K. and Tanaka, K. Independence of Roseman moves including triple points. to appear in Algebr. Geom. Topol. Google Scholar
[14] Kervaire, M. A. Sur l'invariant de Smale d'un plongement (French). Comment. Math. Helv. 34 (1960), 127–139.Google Scholar
[15] Kervaire, M. A. Knot cobordism in codimension two. Manifolds-Amsterdam (1970) (Proc. Nuffic Summer School). Lecture Notes in Math. vol. 197 (Springer, Berlin.), pp. 83–105.Google Scholar
[16] Max, N. and Banchoff, T. Every sphere eversion has a quadruple point. Contributions to Analysis and Geometry (Baltimore, Md., 1980); (Johns Hopkins University Press, Baltimore, Md., 1981), pp. 191–209.Google Scholar
[17] Melikhov, S. A. Sphere eversions and realisation of mappings. Proc. Steklov Inst. Math. 247 (2004), 143–163.Google Scholar
[18] Nowik, T. Quadruple points of regular homotopies of surfaces in 3-manifolds. Topology 39 (2000), 1069–1088.Google Scholar
[19] Nowik, T. Automorphisms and embeddings of surfaces and quadruple points of regular homotopies. J. Differential Geom. 58 (2001), 421–455.Google Scholar
[20] Oshiro, K. and Tanaka, K. On rack colorings for surface-knot diagrams without branch points. Topology Appl. 196 (2015), part B, 921–930.Google Scholar
[21] Roseman, D. Reidemeister-type moves for surfaces in four-dimensional space. Knot theory (Warsaw, 1995), 347–380, Banach Center Publ. 42. (Polish Acad. Sci., Warsaw, 1998).Google Scholar
[22] Rourke, C. and Sanderson, B. The compression theorem I. Geom. Topol. 5 (2001), 399–429.Google Scholar
[23] Satoh, S. Double decker sets of generic surfaces in 3-space as homology classes. Illinois J. Math. 45 (2001), 823–832.Google Scholar
[24] Takase, M. An Ekholm–Szűcs-type formula for codimension one immersions of 3-manifolds up to bordism. Bull. London Math. Soc. 32 (2007), 163–178.Google Scholar