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 , we can define the Arf invariant, and Arf()=0. We prove that, for any -component 1-link , there exists a -component ordinary sense slice 1-link of which is a sublink.

**[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 -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 -curves in 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 -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**

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