## The Michigan Mathematical Journal

### A note on cabled slice knots and reducible surgeries

Jeffrey Meier

#### Article information

Source
Michigan Math. J. Volume 66, Issue 2 (2017), 269-276.

Dates
First available in Project Euclid: 27 March 2017

https://projecteuclid.org/euclid.mmj/1490639817

Digital Object Identifier
doi:10.1307/mmj/1490639817

Zentralblatt MATH identifier
1370.57004

#### Citation

Meier, Jeffrey. A note on cabled slice knots and reducible surgeries. Michigan Math. J. 66 (2017), no. 2, 269--276. doi:10.1307/mmj/1490639817. https://projecteuclid.org/euclid.mmj/1490639817

#### References

• [1] M. Doig and S. Wehrli, A combinatorial proof of the homology cobordism classification of lens spaces, preprint, 2015, arXiv:1505.06970.
• [2] M. Eudave Muñoz, Band sums of links which yield composite links. The cabling conjecture for strongly invertible knots, Trans. Amer. Math. Soc. 330 (1992), no. 2, 463–501.
• [3] D. Gabai, Foliations and surgery on knots, Bull. Amer. Math. Soc. (N.S.) 15 (1986), no. 1, 83–87.
• [4] F. González-Acuña and H. Short, Knot surgery and primeness, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 89–102.
• [5] C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371–415.
• [6] C. Gordon, Dehn surgery and 3-manifolds, Low dimensional topology, IAS/Park City Math. Ser., 15, pp. 21–71, Amer. Math. Soc., Providence, RI, 2009.
• [7] C. McA. Gordon, Dehn surgery and satellite knots, Trans. Amer. Math. Soc. 275 (1983), no. 2, 687–708.
• [8] C. M. Gordon and J. Luecke, Reducible manifolds and Dehn surgery, Topology 35 (1996), no. 2, 385–409.
• [9] J. E. Greene, $L$-space surgeries, genus bounds, and the cabling conjecture, J. Differential Geom. 100 (2015), no. 3, 491–506.
• [10] C. Grove, Cabling Conjecture for small bridge number, preprint, 2015, arXiv:1507.01317.
• [11] C. Hayashi and K. Motegi, Dehn surgery on knots in solid tori creating essential annuli, Trans. Amer. Math. Soc. 349 (1997), no. 12, 4897–4930.
• [12] C. Hayashi and K. Shimokawa, Symmetric knots satisfy the cabling conjecture, Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 3, 501–529.
• [13] M. Hedden, S.-G. Kim, and C. Livingston, Topologically slice knots of smooth concordance order two, J. Differential. Geom. 102 (2016), 353–393.
• [14] J. A. Hoffman, There are no strict great $x$-cycles after a reducing or $P^{2}$ surgery on a knot, J. Knot Theory Ramifications 7 (1998), no. 5, 549–569.
• [15] J. Hom, Bordered Heegaard Floer homology and the tau-invariant of cable knots, J. Topol. 7 (2014), no. 2, 287–326.
• [16] J. Howie, A proof of the Scott-Wiegold conjecture on free products of cyclic groups, J. Pure Appl. Algebra 173 (2002), no. 2, 167–176.
• [17] J. Howie and Can, Dehn surgery yield three connected summands? Groups Geom. Dyn. 4 (2010), no. 4, 785–797.
• [18] W. W. Menasco and M. B. Thistlethwaite, Surfaces with boundary in alternating knot exteriors, J. Reine Angew. Math. 426 (1992), 47–65.
• [19] K. Miyazaki, Nonsimple, ribbon fibered knots, Trans. Amer. Math. Soc. 341 (1994), no. 1, 1–44.
• [20] L. Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), 737–745.
• [21] P. S. Ozsváth and Z. Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003), no. 2, 179–261.
• [22] P. S. Ozsváth and Z. Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), no. 3, 1027–1158.
• [23] P. S. Ozsváth and Z. Szabó, Knot Floer homology and integer surgeries, Algebr. Geom. Topol. 8 (2008), no. 1, 101–153.
• [24] P. S. Ozsváth and Z. Szabó, Knot Floer homology and rational surgeries, Algebr. Geom. Topol. 11 (2011), no. 1, 1–68.
• [25] N. Sayari, Reducible Dehn surgery and the bridge number of a knot, J. Knot Theory Ramifications 18 (2009), no. 4, 493–504.
• [26] L. G. Valdez Sánchez, Dehn fillings of $3$-manifolds and non-persistent tori, Topology Appl. 98 (1999), no. 1–3, 355–370, II Iberoamerican Conference on Topology and Its Applications (Morelia, 1997).
• [27] Y.-Q. Wu, Dehn surgery on arborescent knots, J. Differential Geom. 43 (1996), no. 1, 171–197.
• [28] Z. Wu, A cabling formula for $\upsilon^{+}$ invariant, preprint, 2015, arXiv:1501.04749.
• [29] N. Zufelt, Divisibility of great webs and reducible Dehn surgery, preprint, 2014, arXiv:1410.3442.