Band sums of links which yield composite links. The cabling conjecture for strongly invertible knots
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- by Mario Eudave Muñoz PDF
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Abstract:
We consider composite links obtained by bandings of another link. It is shown that if a banding of a split link yields a composite knot then there is a decomposing sphere crossing the band in one arc, unless there is such a sphere disjoint from the band. We also prove that if a banding of the trivial knot yields a composite knot or link then there is a decomposing sphere crossing the band in one arc. The last theorem implies, via double branched covers, that the only way we can get a reducible manifold by surgery on a strongly invertible knot is when the knot is cabled and the surgery is via the slope of the cabling annulus.References
- Steven A. Bleiler, Prime tangles and composite knots, Knot theory and manifolds (Vancouver, B.C., 1983) Lecture Notes in Math., vol. 1144, Springer, Berlin, 1985, pp. 1–13. MR 823278, DOI 10.1007/BFb0075008
- Steven A. Bleiler, Banding, twisted ribbon knots, and producing reducible manifolds via Dehn surgery, Math. Ann. 286 (1990), no. 4, 679–696. MR 1045396, DOI 10.1007/BF01453596
- Steven A. Bleiler and Mario Eudave Muñoz, Composite ribbon number one knots have two-bridge summands, Trans. Amer. Math. Soc. 321 (1990), no. 1, 231–243. MR 968881, DOI 10.1090/S0002-9947-1990-0968881-9
- Steven Bleiler and Martin Scharlemann, A projective plane in $\textbf {R}^4$ with three critical points is standard. Strongly invertible knots have property $P$, Topology 27 (1988), no. 4, 519–540. MR 976593, DOI 10.1016/0040-9383(88)90030-4
- Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987), no. 2, 237–300. MR 881270, DOI 10.2307/1971311
- Mario Eudave Muñoz, Surgery on strongly invertible knots, An. Inst. Mat. Univ. Nac. Autónoma México 26 (1986), 41–57 (1987) (Spanish). MR 906326
- Mario Eudave Muñoz, Primeness and sums of tangles, Trans. Amer. Math. Soc. 306 (1988), no. 2, 773–790. MR 933317, DOI 10.1090/S0002-9947-1988-0933317-1 —, Band sums of knots and composite links. The cabling conjecture for strongly invertible knots, Ph.D. thesis, UCSB, 1990.
- Francisco González-Acuña and Hamish Short, Knot surgery and primeness, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 89–102. MR 809502, DOI 10.1017/S0305004100063969
- C. McA. Gordon, Dehn surgery and satellite knots, Trans. Amer. Math. Soc. 275 (1983), no. 2, 687–708. MR 682725, DOI 10.1090/S0002-9947-1983-0682725-0
- John Hempel, $3$-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
- Paik Kee Kim and Jeffrey L. Tollefson, Splitting the PL involutions of nonprime $3$-manifolds, Michigan Math. J. 27 (1980), no. 3, 259–274. MR 584691
- W. B. Raymond Lickorish, Prime knots and tangles, Trans. Amer. Math. Soc. 267 (1981), no. 1, 321–332. MR 621991, DOI 10.1090/S0002-9947-1981-0621991-2
- William W. Menasco and Morwen B. Thistlethwaite, Surfaces with boundary in alternating knot exteriors, J. Reine Angew. Math. 426 (1992), 47–65. MR 1155746
- José M. Montesinos, Surgery on links and double branched covers of $S^{3}$, Knots, groups, and $3$-manifolds (Papers dedicated to the memory of R. H. Fox), Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N.J., 1975, pp. 227–259. MR 0380802
- Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
- Martin Scharlemann, Smooth spheres in $\textbf {R}^4$ with four critical points are standard, Invent. Math. 79 (1985), no. 1, 125–141. MR 774532, DOI 10.1007/BF01388659
- Martin G. Scharlemann, Unknotting number one knots are prime, Invent. Math. 82 (1985), no. 1, 37–55. MR 808108, DOI 10.1007/BF01394778
- Martin Scharlemann, Sutured manifolds and generalized Thurston norms, J. Differential Geom. 29 (1989), no. 3, 557–614. MR 992331
- Martin Scharlemann, Producing reducible $3$-manifolds by surgery on a knot, Topology 29 (1990), no. 4, 481–500. MR 1071370, DOI 10.1016/0040-9383(90)90017-E
- Martin Scharlemann and Abigail Thompson, Unknotting number, genus, and companion tori, Math. Ann. 280 (1988), no. 2, 191–205. MR 929535, DOI 10.1007/BF01456051
- Martin Scharlemann and Abigail Thompson, Link genus and the Conway moves, Comment. Math. Helv. 64 (1989), no. 4, 527–535. MR 1022995, DOI 10.1007/BF02564693
- Jeffrey L. Tollefson, Involutions on $S^{1}\times S^{2}$ and other $3$-manifolds, Trans. Amer. Math. Soc. 183 (1973), 139–152. MR 326738, DOI 10.1090/S0002-9947-1973-0326738-0
- Friedhelm Waldhausen, Über Involutionen der $3$-Sphäre, Topology 8 (1969), 81–91 (German). MR 236916, DOI 10.1016/0040-9383(69)90033-0
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 330 (1992), 463-501
- MSC: Primary 57M25; Secondary 57N10
- DOI: https://doi.org/10.1090/S0002-9947-1992-1112545-X
- MathSciNet review: 1112545