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BERGE'S KNOTS IN THE FIBER SURFACES OF GENUS ONE, LENS SPACE AND FRAMED LINKS

YUICHI YAMADA

Department of Systems Engineering, The University of Electro-Communications, 1-5-1,Chofugaoka, Chofu, Tokyo, 182-8585, Japan

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https://doi.org/10.1142/S0218216505003774Cited by:10 (Source: Crossref)

Abstract

In 1990, John Berge described several families of knots in the three-dimensional sphere which have non-trivial Dehn surgeries yielding lens spaces. We study a subfamily of them from the view point of resolution of singularity of complex curves and surfaces, Kirby calculus in topology of four-dimensional manifolds and A'Campo's divide knot theory.

Keywords:

AMSC: Primary 57M25, Secondary 55A25

References

N. A'Campo, Inst. Hautes Etudes Sci. Publ. Math. 88, 151 (1998). Crossref, Google Scholar
N. A'Campo, Inst. Hautes Etudes Sci. Publ. Math. 88, 171 (1998). Crossref, Google Scholar
N. A'Campo, Ann. Fac. Sci. Toulouse 8, 5 (1999). Crossref, Web of Science, Google Scholar
V. I. Arnold , S. M. Gusein-Zade and A. N. Varchenko , Singularities of Differentiable Maps, Volume II , Monographs in Mathematics 83 ( Birkhauser Boston, Inc. , Boston, Massachusetts , 1988 ) . Crossref, Google Scholar
K. Baker, Knots on once-punctured fibers, dissertation, The University of Texas at Austin (2004) . Google Scholar
W. Barth , C. Peters and A. Van de Ven , Compact Complex Surfaces , Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge 4 ( Springer-Verlag , Berlin , 1984 ) . Crossref, Google Scholar
J. Berge, Some knots with surgeries yielding lens spaces, unpublished manuscript (1990) . Google Scholar
O. Couture and B. Perron, J. Knot Theory Ramifications 9, 1 (2000). Link, Web of Science, Google Scholar
R. Fintushel and R. Stern, Math. Z. 175, 33 (1980). Crossref, Web of Science, Google Scholar
H. Goda, M. Hirasawa and Y. Yamada, Tokyo J. Math. 25(2), 485 (2002). Crossref, Web of Science, Google Scholar
R. Gompf and A. Stipsicz , 4-Manifolds and Kirby Calculus , Graduate Studies in Mathematics 20 ( American Mathematical Society , Providence , 1999 ) . Crossref, Google Scholar
S. M. Gusein-Zade, Func. Anal. Appl. 8, 10 (1974). Crossref, Google Scholar
J. Harer, A. Kas and R. Kirby, Handlebody Decompositions of Complex Surfaces, Memoirs of the American Mathematical Society, Vol. 62 (1986) . Google Scholar
M. Hirasawa, Topol. Appl. 121, 287 (2002). Crossref, Web of Science, Google Scholar
R. Kirby , The Topology of 4-Manifolds , Lecture Notes in Mathematics 1374 ( Springer-Verlag , Berlin , 1989 ) . Crossref, Google Scholar
H. Laufer , Normal Two-Dimensional Singularities , Annals of Mathematics Studies 71 ( Princeton Press , 1971 ) . Google Scholar
D. Rolfsen , Knots and Links ( Publish or Perish , Berkeley , 1976 ) . Google Scholar
L. Rudolph, Bull. Amer. Math. Soc. 29, 51 (2001). Crossref, Web of Science, Google Scholar
M. Ue, Algebr. Geom. Topol. 1(26), 549 (2001). Crossref, Google Scholar
Y. Yamada, Finite Dehn surgery along A'Campo's divide knots, to appear in Singular Theory and Its Applications, Proc. 12th MSJ-IRI Symposium . Google Scholar
Y. Yamada, New Methods and Subjects in Singularity Theory, Sūrikaisekikenkyūsho Kōkyūroku 1374, 179 (2004). Google Scholar