Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Some properties of ordinary sense slice 1-links: some answers to problem (26) of Fox


Author: Eiji Ogasa
Journal: Proc. Amer. Math. Soc. 126 (1998), 2175-2182
MSC (1991): Primary 57M25, 57Q45
DOI: https://doi.org/10.1090/S0002-9939-98-04299-3
MathSciNet review: 1443400
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that, for any ordinary sense slice 1-link $L$, we can define the Arf invariant, and Arf($L$)=0. We prove that, for any $m$-component 1-link $L_{1}$, there exists a $3m$-component ordinary sense slice 1-link $L_{2}$ of which $L_{1}$ is a sublink.


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

  • [1] J. S. Carter and M. Saito, Knotted surfaces, braid movies, and beyond, Knots and quantum gravity, edited by J. C. Baez, Clarendon Press, Oxford (1994), 191-229. MR 95m:57036
  • [2] T. D. Cochran and K. Orr, Not all links are concordant to boundary links, Ann. of Math. 138 (1993), 519-554. MR 95c:57042
  • [3] R.H.Fox, Some problems in knot theory, Topology of 3-manifolds and related topics, Proc. 1961 Top. Inst. Georgia, Prentice-Hall, Englewood Cliffs, NJ, 1962, pp. 168-176. MR 25:3523
  • [4] P. Gilmer, Link cobordism in rational homology 3-spheres, J. Knot Theory Ramifications 2 (1993), 285-320. MR 94m:57012
  • [5] P. Gilmer and C. Livingston, The Casson-Gordon invariant and link concordance, Topology 31 (1992), 475-492. MR 93h:57037
  • [6] O. G. Harrold and S. Kinoshita, A theorem on $\theta $-curves and its application to a problem of T. B. Rushing, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), 631-634. MR 83h:57011
  • [7] S. Kamada, A characterization of groups of closed orientable surfaces in 4-space, Topology 33 (1994), 113-122. MR 95a:57002
  • [8] A. Kawauchi, On the Robertello invariants of proper links, Osaka J. Math. 21 (1984), 81-90. MR 85j:57004
  • [9] A. Kawauchi, T. Shibuya and S. Suzuki, Descriptions on surfaces in four-space, I. Normal forms, Math Sem. Notes Kobe Univ. 10 (1982), 75-125 II. Singularities and cross-sectional links Math sem. notes Kobe Univ. 11 (1983), 31-69. MR 84d:57017; MR 85j:57033
  • [10] M. Kervaire, Les noeudes de dimensions supérieures, Bull.Soc.Math.France 93 (1965), 225-271. MR 32:6479
  • [11] S. Kinoshita, On $\theta _{n}$-curves in $\mathbb{R}^{3}$ and their constituent knots, In:Topology and Computer sciences, Kinokuniya, Tokyo (1987), 211-216. MR 92h:57010
  • [12] S. Kinoshita and H. Terasaka, On unions of knots, Osaka J. Math 9 (1957), 131-153. MR 20:4846
  • [13] J. Levine, Doubly sliced knots and doubled disc knots, Michigan Math J. 30 (1983), 249-256. MR 85h:57024
  • [14] J. Levine, Link invariants via the eta-invariant, Comment. Math. Helv. 69 (1994), 82-119. MR 95a:57009
  • [15] K. Miyazaki and A. Yasuhara, Generalized $\sharp $-unknotting operations, J. Math. Soc. Japan 49 (1997), 107-125. MR 97i:57007
  • [16] K. Murasugi, On a certain numerical invariant of link types, TransAMS 117 (1965), 387-422. MR 30:1506
  • [17] E. Ogasa, On the intersection of spheres in a sphere I, University of Tokyo preprint (1995).
  • [18] E. Ogasa, On the intersection of spheres in a sphere II:High dimensional case, University of Tokyo preprint (1995).
  • [19] R.A. Robertello, An invariant of knot cobordism, Comm. Pure. Appl. Math. 18 (1965), 543-555. MR 32:447
  • [20] D. Ruberman, Doubly slice knots and the Casson-Gordon invariants, Trans. Amer. Math. Soc. 279 (1983), 569-588. MR 85e:57025
  • [21] D. Ruberman, The Casson-Gordon invariants in high-dimensional knot theory, Trans. Amer. Math. Soc. 306 (1988), 579-595. MR 89g:57031
  • [22] D. W. Sumners, Invertible knot cobordisms, Comment. Math. Helv. 46 (1971), 240-256. MR 44:7535
  • [23] S. Suzuki, Local knots of 2-spheres in 4-manifolds, Proc. Japan Acad. Sci. 45 (1969), 34-38. MR 40:2101
  • [24] S. Suzuki, Knotting problems of 2-spheres in the 4-sphere, Math. Sem. Notes Kobe Univ. 4 (1976), 241-371. MR 56:3848
  • [25] M. Yamamoto, Knots in spatial embeddings of the complete graph on four vertices, Topology Appl. 36 (1990), 291-298. MR 91m:57007

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 57M25, 57Q45

Retrieve articles in all journals with MSC (1991): 57M25, 57Q45


Additional Information

Eiji Ogasa
Affiliation: Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153, Japan
Email: ogasa@ms.u-tokyo.ac.jp, ogasa@ms513red.ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-98-04299-3
Keywords: Ordinary sense slice 1-links, Arf invariants, $n$-dimensional knots and links, Suzuki-Terasaka diagrams, realizable 4-tuple of links
Received by editor(s): April 10, 1996
Received by editor(s) in revised form: December 27, 1996
Additional Notes: This research was partially supported by Research Fellowships of the Promotion of Science for Young Scientists.
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society