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Annulus twist and diffeomorphic 4-manifolds

Published online by Cambridge University Press:  17 May 2013

TETSUYA ABE ,
IN DAE JONG ,
YUKA OMAE  and
MASANORI TAKEUCHI
TETSUYA ABE
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan. e-mail: [email protected]
IN DAE JONG
Affiliation:
Faculty of Liberal Arts and Sciences, Osaka Prefecture University, 1-1 Gakuen-cho, Nakaku, Sakai, Osaka 599-8531, Japan. e-mail: [email protected]
YUKA OMAE
Affiliation:
Osaka Prefectural Kitano High School, Osaka 532-0025, Japan. e-mail: [email protected]
MASANORI TAKEUCHI
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan.
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Abstract

We give a method for obtaining infinitely many framed knots which represent a diffeomorphic 4-manifold. We also study a relationship between the n-shake genus and the 4-ball genus of a knot. Furthermore we give a construction of homotopy 4-spheres from a slice knot with unknotting number one.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 2 , September 2013 , pp. 219 - 235
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

REFERENCES

[1]Abe, T., Hanaki, R. and Higa, R.The unknotting number and band-unknotting number of a knot. Osaka J. Math. 49 (2012), no. 2, 523–550.Google Scholar
[2]Abe, T. and Kanenobu, T. Unoriented band-surgery on knots and links, preprint.Google Scholar
[3]Akbulut, S.Knots and exotic smooth structures on 4-manifolds. J. Knot Theory Ramifications 2 (1993), no. 1, 1–10.CrossRefGoogle Scholar
[4]Akbulut, S.On 2-dimensional homology classes of 4-manifolds. Math. Proc. Camb. Phil. Soc. 82 (1977), no. 1, 99–106.CrossRefGoogle Scholar
[5]Akbulut, S. 4-manifolds, draft of a book (2012), available at http://www.math.msu.edu/~akbulut/papers/akbulut.lec.pdfGoogle Scholar
[6]Brakes, R.Manifolds with multiple knot-surgery descriptions. Math. Proc. Camb. Phil. Soc. 87 (1980), no. 3, 443–448.CrossRefGoogle Scholar
[7]Cerf, J.Sur les diffeomorphismes de la sphere de dimension trois (Γ4=0). Lecture Notes in Math. No. 53 (Springer-Verlag, 1968), xii+133 pp.Google Scholar
[8]Fox, R. H. and Milnor, J. W.Singularities of 2-spheres in 4-space and cobordism of knots. Osaka J. Math. 3 (1996), 257–267.Google Scholar
[9]Gabai, D.Foliations and the topology of 3-manifolds, III. J. Differential Geom. 26 (1987), no. 3, 479–536.Google Scholar
[10]Gompf, R. and Miyazaki, K.Some well-disguised ribbon knots. Topology Appl. 64 (1995), no. 2, 117–131.CrossRefGoogle Scholar
[11]Gompf, R. and Stipsicz, A.4-manifolds and Kirby calculus. Graduate Studies in Math. 20 (Amer. Math. Soc. 1999), xvi+558.Google Scholar
[12]Kawauchi, A.Mutative hyperbolic homology 3-spheres with the same Floer homology. Geom. Dedicata 61 (1996), no. 2, 205–217.CrossRefGoogle Scholar
[13]Kirby, R.Problems in low-dimensional topology. AMS/IP Stud. Adv. Math. 2 (2), Geometric topology (Athens, GA, 1993), 35–473 (Amer. Math. Soc. 1997).Google Scholar
[14]Kouno, R. 3–manifold with infinitely many knot surgery descriptions (in Japanese) Master's thesis, Nihon University (2002).Google Scholar
[15]Neumann, W. and Zagier, D.Volumes of hyperbolic three-manifolds. Topology 24 (1985), no. 3, 307–332.CrossRefGoogle Scholar
[16]Lickorish, W. B. R.Shake-slice knots. Lecture Notes in Math. 722 (1979), 67–70.CrossRefGoogle Scholar
[17]Lickorish, W. B. R.Surgery on knots. Proc. Amer. Math. Soc. 60 (1976), 296–298.CrossRefGoogle Scholar
[18]Livingston, C.More 3-manifolds with multiple knot-surgery and branched-cover descriptions. Math. Proc. Camb. Phil. Soc. 91 (1982), no. 3, 473–475.CrossRefGoogle Scholar
[19]Omae, Y. 4-manifolds constructed from a knot and the shake genus (in Japanese). Master's thesis, Osaka University (2011).Google Scholar
[20]Osoinach, J.Manifolds obtained by surgery on an infinite number of knots in S 3. Topology 45 (2006), no. 4, 725–733.CrossRefGoogle Scholar
[21]Saito, T. and Teragaito, M.Knots yielding diffeomorphic lens spaces by Dehn surgery. Pacific J. Math. 244 (2010), no. 1, 169–192.CrossRefGoogle Scholar
[22]Takeuchi, M. Infinitely many distinct framed knots which represent a diffeomorphic 4-manifold. (in Japanese) Master thesis, Osaka University (2009).Google Scholar
[23]Teragaito, M.A Seifert fibered manifold with infinitely many knot-surgery descriptions, Int. Math. Res. Not. Int. Math. Res. Not. no. 9 (2007), Art. ID rnm 028, 16 pp.Google Scholar
[24]Teragaito, M.Homology handles with multiple knot-surgery descriptions. Topology Appl. 56 (1994), no. 3, 249–257.CrossRefGoogle Scholar
[25]Terasaka, H.On null-equivalent knots. Osaka Math. J. 11 (1959), 95–113.Google Scholar
[26]Winter, B. On codimension two ribbon embeddings. arXiv:0904.0684 (2009).Google Scholar