There are knots whose tunnel numbers go down under connected sum
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- by Kanji Morimoto
- Proc. Amer. Math. Soc. 123 (1995), 3527-3532
- DOI: https://doi.org/10.1090/S0002-9939-1995-1317043-4
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Abstract:
In this paper, we show that there are infinitely many tunnel number two knots K such that the tunnel number of $K\# K’$ is equal to two again for any 2-bridge knot $K’$.References
- Joan S. Birman and Hugh M. Hilden, Heegaard splittings of branched coverings of $S^{3}$, Trans. Amer. Math. Soc. 213 (1975), 315–352. MR 380765, DOI 10.1090/S0002-9947-1975-0380765-8
- Michel Boileau, Markus Rost, and Heiner Zieschang, On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces, Math. Ann. 279 (1988), no. 3, 553–581. MR 922434, DOI 10.1007/BF01456287 F. Bonahon and L. Seibemann, Geometric splittings of knots, and Conway’s algebraic knots, preprint.
- William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450, DOI 10.1090/cbms/043
- William H. Jaco and Peter B. Shalen, Seifert fibered spaces in $3$-manifolds, Mem. Amer. Math. Soc. 21 (1979), no. 220, viii+192. MR 539411, DOI 10.1090/memo/0220
- Klaus Johannson, Homotopy equivalences of $3$-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR 551744, DOI 10.1007/BFb0085406
- Kanji Morimoto, On the additivity of tunnel number of knots, Topology Appl. 53 (1993), no. 1, 37–66. MR 1243869, DOI 10.1016/0166-8641(93)90099-Y
- Kanji Morimoto, Characterization of tunnel number two knots which have the property “$2+1=2$”, Topology Appl. 64 (1995), no. 2, 165–176. MR 1340868, DOI 10.1016/0166-8641(94)00096-L
- Kanji Morimoto and Makoto Sakuma, On unknotting tunnels for knots, Math. Ann. 289 (1991), no. 1, 143–167. MR 1087243, DOI 10.1007/BF01446565
- F. H. Norwood, Every two-generator knot is prime, Proc. Amer. Math. Soc. 86 (1982), no. 1, 143–147. MR 663884, DOI 10.1090/S0002-9939-1982-0663884-7
- Martin Scharlemann, Tunnel number one knots satisfy the Poenaru conjecture, Topology Appl. 18 (1984), no. 2-3, 235–258. MR 769294, DOI 10.1016/0166-8641(84)90013-0
- H. Seifert, Topologie Dreidimensionaler Gefaserter Räume, Acta Math. 60 (1933), no. 1, 147–238 (German). MR 1555366, DOI 10.1007/BF02398271
- O. Ja. Viro, Links, two-sheeted branching coverings and braids, Mat. Sb. (N.S.) 87(129) (1972), 216–228 (Russian). MR 0298649
Bibliographic Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 3527-3532
- MSC: Primary 57M25
- DOI: https://doi.org/10.1090/S0002-9939-1995-1317043-4
- MathSciNet review: 1317043