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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]
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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 

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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