3 . , , , ( [13]). , [1] , ). , r 4 K ( r ) , r ( , ? ,Gordon–Luecke[2] , K , , , , . , 3 . , , , .[10] ( [10] ). • (toroidal3-manifold) • (Seifertﬁbered3-manifold) • (reducible) ) , (GeometrizationConjecture) , ( (Thurston ,[15]). ) , , , ( − 1 K ( r ) .

(1)

KLEIN BOTTLAL SURGERY AND GENERA OF KNOTS

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Abstract. In this article, we study the creation of Klein bottles by surgery on knots in the 3-sphere. We give an upper bound for the slopes yielding Klein bottles in terms of the genera of knots.

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. References

1. I. Agol,Bounds on exceptional Dehn filling, Geom. Topol.4(2000), 431–449.

2. C. McA. Gordon and J. Luecke,Dehn surgeries on knots creating essential tori, I, Comm. Anal. Geom.

3(1995), 597–644.

3. H. Goda and M. Teragaito,Dehn surgeries on knots which yield lens spaces and genera of knots, Math.

Proc. Cambridge Philos. Soc.129(2000), no. 3, 501–515.

4. C. Hayashi and K. Motegi,Only single twists on unknots can produce composite knots, Trans. Amer.

Math. Soc.349(1997), 4465–4479.

5. K. Ichihara,Exceptional surgeries and genera of knots, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 4, 66–67.

6. , Boundary slopes of non-orientable Seifert surfaces for knots, (^"!# ,

$&%'&(*)+

,.-/0#21435647*8 ),^7*89*: ‘Art of Low Dimensional Topology VII’^;<9 (^=>?

@BACDEGF

, 2001.2.15–17).

7. K. Ichihara, M. Ohtouge and M. Teragaito, Boundary slopes of non-orientable Seifert surfaces for knots, to appear in Top. Appl. .

8. K. Ichihara and M. Teragaito,Klein bottlal surgery and genera of knots, preprint.

9. S. Kamada,Nonorientable surfaces in 4-space, Osaka J. Math.26(1989), 367–385.

10. J. Luecke, Dehn surgery on knots in the 3-sphere, Proceedings of the International Congress of Math- ematicians, Vol. 1, 2 (Zurich, 1994), 585–594, Birkhauser, Basel, 1995.

11. D. Rolfsen, Knots and links, Mathematics Lecture Series, No. 7. Publish or Perish, Inc., Berkeley, Calif., 1976.

12. H. Schubert,Knoten und Vollringe, Acta Math.90(1953), 131–286

13. M. Teragaito, Creating Klein bottles by surgery on knots, J. Knot Theory Ramifications 10(2001), 781–794.

14. M. Teragaito,Dehn surgeries on composite knots creating Klein bottles, J. Knot Theory Ramifications 8(1999), 391–395.

15. W. P. Thurston, The geometry and topology of 3-manifolds, Lecture notes, Princeton University, 1978.

152–8552HIJ2KMLN4OP 2–12–1QRHI&SUTVUWXY&SUG78Z[Y\^]_Z`a"b . Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo 152-8552, Japan

E-mail address:ichihara@is.titech.ac.jp

739–8524cd&ef"g.h 1–1–1,d&e"i`i`VUjk`lmZ[`jk`no . Department of Mathematics and Mathematics Education, Faculty of Education, Hiroshima University, 1-1-1 Kagamiyama, Higashi-hiroshima 739-8524, Japan

E-mail address:teragai@hiroshima-u.ac.jp