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Banach Center Publications

1998 | 42 | 1 | 347-380
Tytuł artykułu

Reidemeister-type moves for surfaces in four-dimensional space

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We consider smooth knottings of compact (not necessarily orientable) n-dimensional manifolds in $ℝ^{n+2}$ (or $S^{n+2}$), for the cases n=2 or n=3. In a previous paper we have generalized the notion of the Reidemeister moves of classical knot theory. In this paper we examine in more detail the above mentioned dimensions. Examples are given; in particular we examine projections of twist-spun knots. Knot moves are given which demonstrate the triviality of the 1-twist spun trefoil. Another application is a smooth version of a result of Homma and Nagase on a set of moves for regular homotopies of surfaces.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
347-380
Opis fizyczny
Daty
wydano
1998
Twórcy
autor
• The University of Iowa, Iowa City, Iowa, U.S.A.
Bibliografia
• [C-S] J. S. Carter and M. Saito, Reidemeister moves for surface isotopies and their interpretation as moves to movies, J. Knot Theory Ramifications (2) (1993), 251-284.
• [H1] A. J. Hanson, videotape entitled Knot⁴, exhibited in Small Animation Theater o SIGGRAPH 93, Anaheim, CA, August 1-8, 1993. Published in Siggraph Video Review 93, Scene 1 (1993).
• [H2] MeshView 4D, a 4d surface viewer for meshes for SGI machines, available via ftp from the Geometry Center (1994).
• [RS5] D. Roseman, Design of a mathematicians' drawing program, in: Computer Graphics Using Object-Oriented Programming, S. Cunningham, J. Brown, N. Craghill and M. Fong (eds.), John Wiley & Sons, 1992, 279-296.
• [RS6] D. Roseman, Motions of flexible objects, in: Modern Geometric Computing for Visualization, T. L. Kunii and Y. Shinagawa (eds.), Springer, 1992, 91-120.
• [RS7] D. Roseman (with D. Mayer), Viewing knotted spheres in 4-space, video (8 mins.), produced at the Geometry Center, June 1992.
• [RS8] D. Roseman (with D. Mayer and O. Holt), Twisting and turning in 4 dimensions, video (19 mins.), produced at the Geometry Center, August 1993, distributed by Great Media, Nicassio, CA.
• [RS9] D. Roseman (with D. Mayer and O. Holt), Unraveling in 4 dimensions, video (18 mins.), produced at the Geometry Center, July 1994, distributed by Great Media, Nicassio, CA.
• [F] G. K. Francis, A Topological Picturebook, Springer, 1987.
• [GL] C. Giller, Towards a classical knot theory for surfaces in $ℝ^4$, Illinois J. Math. 26 (1982), 591-631.
• [GR] C. Gordon, Some aspects of classical knot theory, in: Knot Theory, Lecture Notes in Math. 685, Springer.
• [HM-NG] T. Homma and T. Nagase, On elementary deformations of maps of surfaces into 3-manifolds, in: Topology and Computer Science, Kinokuniya Co. Ltd., Tokyo, 1987, 1-20.
• [HR1] M. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242-276.
• [HR2] M. Hirsch, Differential Topology, Grad. Texts in Math. 33, Springer, 1976.
• [MR] B. Morin, Formes canoniques des singularités d'une application différentiable, C. R. Acad. Sci. Paris 26 (1965), 5662-5665.
• [RH] V. A. Rohlin, The embedding of non-orientable three-manifolds into five-dimensional Euclidean space, Dokl. Akad. Nauk SSSR 160 (1965), 549-551 (in Russian; English transl.: Soviet Math. Dokl. 6 (1965), 153-156).
• [RD] K. Reidemeister, Knotentheorie, Springer, 1932; reprint: 1974.
• [RS1] D. Roseman, The spun square knot is the spun granny knot, Bol. Soc. Math. Mex. (1975), 49-55.
• [RS2] D. Roseman, Spinning knots about submanifolds; spinning knots about projections of knots, Topology Appl. 31 (1989), 225-241,
• [RS3] D. Roseman, Projections of codimension two embeddings, to appear.
• [RS4] D. Roseman, Elementary moves for higher dimensional knots, preprint.
• [W] C. T. C. Wall, All 3-manifolds imbed in 4-space, Bull. Amer. Math. Soc. 71 (1965), 564-567.
• [ZM] E. C. Zeeman, Twisting spin knots, Trans. Amer. Math. Soc. 115 (1965), 471-495.
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Bibliografia
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