Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Towards invariants of surfaces in $ 4$-space via classical link invariants

Author: Sang Youl Lee
Journal: Trans. Amer. Math. Soc. 361 (2009), 237-265
MSC (2000): Primary 57Q45; Secondary 57M25
Published electronically: August 13, 2008
MathSciNet review: 2439406
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we introduce a method to construct ambient isotopy invariants for smooth imbeddings of closed surfaces into $ 4$-space by using hyperbolic splittings of the imbedded surfaces and an arbitrary given isotopy or regular isotopy invariant of classical knots and links in $ 3$-space. Using this construction, adopting the Kauffman bracket polynomial as an example, we produce some invariants.

References [Enhancements On Off] (What's this?)

  • 1. M. Asada, An unknotting sequence for surface-knots represented by ch-diagrams and their genera, Kobe J. Math. 18 (2001), 163-180. MR 1907671 (2003d:57049)
  • 2. J.S. Carter, J.H. Rieger, M. Saito, A combinatorial description of knotted surfaces and their isotopies, Adv. Math. 127(1997), 1-51. MR 1445361 (98c:57023)
  • 3. J.S. Carter, D. Jelsovsky, S. Kamada, L. Langford, M. Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc. 355 (2003), 3947-3989. MR 1990571 (2005b:57048)
  • 4. D. Cox, J. Little, D. O'Shea, Ideals, Varieties, and Algorithms, Springer, 1997. MR 1417938 (97h:13024)
  • 5. J.S. Carter, M. Saito, Knotted surfaces and their diagrams, Mathematical surveys and monographs 55, Amer. Math. Soc., Providence, RI, 1998. MR 1487374 (98m:57027)
  • 6. R.H. Fox, A quick trip through knot theory, in Toplogy of $ 3$-manifolds and Related Topics, Prentice-Hall, Inc., Englewood Cliffs, N.J.(1962), 120-167. MR 0140099 (25:3522)
  • 7. R.H. Fox and J.W. Milnor, Singularities of $ 2$-spheres in $ 4$-space and equivalence of knots, (unpublished version).
  • 8. S. Kamada, Non-orientable surfaces in $ 4$-space, Osaka J. Math. 26(1989), 367-385. MR 1017592 (91g:57022)
  • 9. S. Kamada, Surfaces in $ \mathbb{R}^4$ of braid index three are ribbon, J. Knot Theory Ramifications 1(1992), 137-160. MR 1164113 (93h:57039)
  • 10. S. Kamada, A characterization of groups of closed orientable surfaces in $ 4$-space, Topology 33(1994), 113-122. MR 1259518 (95a:57002)
  • 11. S. Kamada, Braid and Knot Theory in Dimension Four, Mathematical Surveys and Monographs 95(2002), American Mathematical Society. MR 1900979 (2003d:57050)
  • 12. S. Kamada, in preparation.
  • 13. L.H. Kauffman, State models and the Jones polynomial, Topology 26(1987), 395-407. MR 899057 (88f:57006)
  • 14. A. Kawauchi, T. Shibuya, S. Suzuki, Descriptions on surfaces in four-space, I; Normal forms, Math. Sem. Notes Kobe Univ. 10(1982), 75-125. MR 672939 (84d:57017)
  • 15. A. Kawauchi, A survey of knot theory, Birkhäuser, 1996. MR 1417494 (97k:57011)
  • 16. S. Y. Lee, Invariants of surface links in $ \mathbb{R}^4$ via skein relation, J. Knot Theory Ramifications 17(2008), 439-469.
  • 17. S. Y. Lee, Invariants of surfaces in $ 4$-space via an elementary classical link invariant, preprint available at$ ^\sim$knot.
  • 18. S. Y. Lee, Invariants of oriented surfaces in $ 4$-space via invariants for magnetic graphs, in preparation.
  • 19. L.Tr. Lomonaco, The homotopy groups of knots I. How to compute the algebraic $ 2$-type, Pacific J. Math. 95(1981), 349-390. MR 632192 (83a:57025)
  • 20. Y. Miyazawa, Magnetic graphs and an invariant for virtual links, J. Knot Theory Ramifications 15 (2006), 1319-1334. MR 2286126 (2007k:57014)
  • 21. Y. Miyazawa, A multi-variable polynomial invariant for virtual knots and links, to appear in J. Knot Theory Ramifications.
  • 22. Y. Nakanishi, M. Teragaito, $ 2$-knots from a view of moving picture, Kobe J. Math. 8(1991), 161-172. MR 1159867 (94d:57052)
  • 23. D. Roseman, Reidemeister-type moves for surfaces in four-dimensional space, Knot theory (Warsaw, 1995), Polish Acad. Sci.(1998), 347-380. MR 1634466 (99f:57029)
  • 24. L. Rudolph, Braided surfaces and Seifert ribbons for closed braids, Comm. Math. Helv. 58(1983), 1-37. MR 699004 (84j:57006)
  • 25. M. Soma, Surface-links with square-type ch-graphs, Proceedings of the First Joint Japan-Mexico Meeting in Topology (Morelia, 1999), Topology Appl. 121 (2002), 231-246. MR 1903693 (2003c:57024)
  • 26. F.J. Swenton, On a calculus for $ 2$-knots and surfaces in $ 4$-space, J. Knot Theory Ramifications 10(2001), 1133-1141. MR 1871221 (2002j:57043)
  • 27. K. Yoshikawa, An enumeration of surfaces in four-space, Osaka J. Math. 31(1994), 497-522. MR 1309400 (95m:57037)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57Q45, 57M25

Retrieve articles in all journals with MSC (2000): 57Q45, 57M25

Additional Information

Sang Youl Lee
Affiliation: Department of Mathematics, Pusan National University, Pusan 609-735, Korea

Keywords: Kauffman bracket polynomial, knotted surface, knot with bands, surface link, Yoshikawa moves, ch-diagram
Received by editor(s): December 18, 2006
Published electronically: August 13, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society