Reidemeister-type moves for surfaces in four-dimensional space

Dennis Roseman

Banach Center Publications (1998)

  • Volume: 42, Issue: 1, page 347-380
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Roseman, Dennis. "Reidemeister-type moves for surfaces in four-dimensional space." Banach Center Publications 42.1 (1998): 347-380. <http://eudml.org/doc/208817>.

@article{Roseman1998,
abstract = {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.},
author = {Roseman, Dennis},
journal = {Banach Center Publications},
keywords = {knotted 3-manifolds; regular isotopy; twist-spun knots; knot moves; immersions of surfaces; knotted surfaces; Reidemeister move; twist-spun knot; embedding; isotopy; projection},
language = {eng},
number = {1},
pages = {347-380},
title = {Reidemeister-type moves for surfaces in four-dimensional space},
url = {http://eudml.org/doc/208817},
volume = {42},
year = {1998},
}

TY - JOUR
AU - Roseman, Dennis
TI - Reidemeister-type moves for surfaces in four-dimensional space
JO - Banach Center Publications
PY - 1998
VL - 42
IS - 1
SP - 347
EP - 380
AB - 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.
LA - eng
KW - knotted 3-manifolds; regular isotopy; twist-spun knots; knot moves; immersions of surfaces; knotted surfaces; Reidemeister move; twist-spun knot; embedding; isotopy; projection
UR - http://eudml.org/doc/208817
ER -

References

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  1. [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. Zbl0808.57020
  2. [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). 
  3. [H2] MeshView 4D, a 4d surface viewer for meshes for SGI machines, available via ftp from the Geometry Center (1994). 
  4. [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. 
  5. [RS6] D. Roseman, Motions of flexible objects, in: Modern Geometric Computing for Visualization, T. L. Kunii and Y. Shinagawa (eds.), Springer, 1992, 91-120. 
  6. [RS7] D. Roseman (with D. Mayer), Viewing knotted spheres in 4-space, video (8 mins.), produced at the Geometry Center, June 1992. 
  7. [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. 
  8. [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. 
  9. [F] G. K. Francis, A Topological Picturebook, Springer, 1987. 
  10. [GL] C. Giller, Towards a classical knot theory for surfaces in ℝ 4 , Illinois J. Math. 26 (1982), 591-631. Zbl0476.57009
  11. [GR] C. Gordon, Some aspects of classical knot theory, in: Knot Theory, Lecture Notes in Math. 685, Springer. Zbl0386.57002
  12. [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. 
  13. [HR1] M. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242-276. Zbl0113.17202
  14. [HR2] M. Hirsch, Differential Topology, Grad. Texts in Math. 33, Springer, 1976. 
  15. [MR] B. Morin, Formes canoniques des singularités d'une application différentiable, C. R. Acad. Sci. Paris 26 (1965), 5662-5665. Zbl0178.26801
  16. [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). 
  17. [RD] K. Reidemeister, Knotentheorie, Springer, 1932; reprint: 1974. 
  18. [RS1] D. Roseman, The spun square knot is the spun granny knot, Bol. Soc. Math. Mex. (1975), 49-55. Zbl0408.57018
  19. [RS2] D. Roseman, Spinning knots about submanifolds; spinning knots about projections of knots, Topology Appl. 31 (1989), 225-241, Zbl0683.57010
  20. [RS3] D. Roseman, Projections of codimension two embeddings, to appear. Zbl0973.57011
  21. [RS4] D. Roseman, Elementary moves for higher dimensional knots, preprint. Zbl1069.57014
  22. [W] C. T. C. Wall, All 3-manifolds imbed in 4-space, Bull. Amer. Math. Soc. 71 (1965), 564-567. Zbl0135.41603
  23. [ZM] E. C. Zeeman, Twisting spin knots, Trans. Amer. Math. Soc. 115 (1965), 471-495. Zbl0134.42902

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