On preimage knots in 𝑆³

Soma, Teruhiko

doi:10.1090/S0002-9939-1987-0891169-7

On preimage knots in $S\sp 3$

Author: Teruhiko Soma
Journal: Proc. Amer. Math. Soc. 100 (1987), 589-592
MSC: Primary 57M25
DOI: https://doi.org/10.1090/S0002-9939-1987-0891169-7
MathSciNet review: 891169
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In [1], Gordon proved that a certain sequence of preimage knots in ${S^3}$ is finite. In this paper, we prove that every sequence of preimage knots is finite by using the Gromov invariant for knots.

[1] C. Gordon, Dehn surgery and satellite knots, Trans. Amer. Math. Soc. 275 (1983), 687-708. MR 682725 (84d:57003)
[2] M. Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1983), 213-307. MR 686042 (84h:53053)
[3] W. Jaco, Lectures on three-manifold topology, C.B.M.S. Regional Conf. Ser. in Math., no. 43, Amer. Math. Soc., Providence, R. I., 1980. MR 565450 (81k:57009)
[4] W. Jaco and P. Shalen, Seifert fibered spaces in -manifolds, Mem. Amer. Math. Soc., No. 220 (1979). MR 539411 (81c:57010)
[5] K. Johanson, Homotopy equivalences of -manifolds with boundaries, Lecture Notes in Math., vol. 761, Springer-Verlag, 1979.
[6] M. Kouno, On knots with companions, Kobe J. Math. 2 (1985), 143-148. MR 847181 (88b:57011)
[7] J. Morgan, On Thurston's uniformization theorem for three-dimensional manifolds, The Smith Conjecture (edited by J. Morgan and H. Bass), Academic Press, 1984, pp. 37-125. MR 758464
[8] T. Soma, The Gromov invariant of links, Invent. Math. 64 (1981), 445-454. MR 632984 (83a:57014)
[9] W. Thurston, The geometry and topology of -manifolds, Mimeographed notes, Princeton Univ., Princeton, N. J., 1977/78. MR 662424 (83j:57006)