journal article
Mar 26, 2026
Surgeries on torus knots, rational balls, and cabling
Abstract
We classify which positive integral surgeries on positive torus knots bound rational homology balls. Additionally, for a given knot
K
we consider which cables
K
p
,
q
admit integral surgeries that bound rational homology balls. For such cables, let
𝒮
(
K
)
be the set of corresponding rational numbers
q
p
. We show that
𝒮
(
K
)
is bounded for each
K
. Moreover, if
n
-surgery on
K
bounds a rational homology ball then
n
is an accumulation point for
𝒮
(
K
)
.
K
we consider which cables
K
p
,
q
admit integral surgeries that bound rational homology balls. For such cables, let
𝒮
(
K
)
be the set of corresponding rational numbers
q
p
. We show that
𝒮
(
K
)
is bounded for each
K
. Moreover, if
n
-surgery on
K
bounds a rational homology ball then
n
is an accumulation point for
𝒮
(
K
)
.
Topics
No keywords indexed for this article. Browse by subject →
References
50
[1]
[1] Aceto, Paolo Rational homology cobordisms of plumbed 3-manifolds, Algebr. Geom. Topol., Volume 20 (2020) no. 3, pp. 1073-1126
10.2140/agt.2020.20.1073
[2]
[2] Aceto, Paolo; Golla, Marco Dehn surgeries and rational homology balls, Algebr. Geom. Topol., Volume 17 (2017) no. 1, pp. 487-527
10.2140/agt.2017.17.487
[3]
[3] Aceto, Paolo; Larson, Kyle Knot concordance and homology sphere groups, Int. Math. Res. Not., Volume 2018 (2018) no. 23, pp. 7318-7334
10.1093/imrn/rnx091
[4]
[4] Akbulut, Selman; Larson, Kyle Brieskorn spheres bounding rational balls, Proc. Am. Math. Soc., Volume 146 (2018) no. 4, pp. 1817-1824
10.1090/proc/13828
[5]
[5] Baker, Kenneth L.; Buck, Dorothy; Lecuona, Ana G. Some knots in S 1 ×S 2 with lens space surgeries, Commun. Anal. Geom., Volume 24 (2016) no. 3, pp. 431-470
10.4310/cag.2016.v24.n3.a1
[6]
[6] Bedient, Richard E. Double branched covers and pretzel knots, Pac. J. Math., Volume 112 (1984) no. 2, pp. 265-272
10.2140/pjm.1984.112.265
[7]
[7] Fernández de Bobadilla, Javier; Luengo, Ignacio; Melle Hernández, Alejandro; Némethi, Andras Classification of rational unicuspidal projective curves whose singularities have one Puiseux pair, Real and complex singularities (Trends in Mathematics), Birkhäuser, 2007, pp. 31-45
10.1007/978-3-7643-7776-2_4
[8]
[8] Borodzik, Maciej; Livingston, Charles Heegaard Floer homology and rational cuspidal curves, Forum Math. Sigma, Volume 2 (2014), e28, 23 pages
10.1017/fms.2014.28
[9]
[9] Bryant, Kathryn Slice implies mutant ribbon for odd 5-stranded pretzel knots, Algebr. Geom. Topol., Volume 17 (2017) no. 6, pp. 3621-3664
10.2140/agt.2017.17.3621
[10]
[10] Burde, Gerhard; Zieschang, Heiner; Heusener, Michael Knots, De Gruyter Studies in Mathematics, 5, Walter de Gruyter, 2014
[11]
[11] Casson, Andrew J.; Harer, John L. Some homology lens spaces which bound rational homology balls, Pac. J. Math., Volume 96 (1981) no. 1, pp. 23-36
10.2140/pjm.1981.96.23
[12]
[12] Cochran, Tim D.; Franklin, Bridget D.; Hedden, Matthew; Horn, Peter D. Knot concordance and homology cobordism, Proc. Am. Math. Soc., Volume 141 (2013) no. 6, pp. 2193-2208
10.1090/s0002-9939-2013-11471-1
[13]
[13] Donald, Andrew; Owens, Brendan Concordance groups of links, Algebr. Geom. Topol., Volume 12 (2012) no. 4, pp. 2069-2093
10.2140/agt.2012.12.2069
[14]
[14] Donaldson, Simon K. An application of gauge theory to four-dimensional topology, J. Differ. Geom., Volume 18 (1983) no. 2, pp. 279-315
10.4310/jdg/1214437665
[15]
[15] Donaldson, Simon K. The orientation of Yang-Mills moduli spaces and 4-manifold topology, J. Differ. Geom., Volume 26 (1987) no. 3, pp. 397-428
10.4310/jdg/1214441485
[16]
[16] Fintushel, Ronald; Stern, Ronald J. A μ-invariant one homology 3-sphere that bounds an orientable rational ball, Four-manifold theory (Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference in the Mathematical Sciences on Four-manifold Theory held at Durham, New Hampshire, July 4–10, 1982) (Contemporary Mathematics), Volume 35, American Mathematical Society (1984), pp. 265-268
10.1090/conm/035
[17]
[17] Greene, Joshua E. The lens space realization problem, Ann. Math. (2), Volume 177 (2013) no. 2, pp. 449-511
10.4007/annals.2013.177.2.3
[18]
[18] Greene, Joshua E. A note on applications of the d-invariant and Donaldson’s theorem, J. Knot Theory Ramifications, Volume 26 (2017) no. 2, 1740006, 8 pages
10.1142/s0218216517400065
[19]
[19] Greene, Joshua E.; Jabuka, Stanislav The slice-ribbon conjecture for 3-stranded pretzel knots, Am. J. Math., Volume 133 (2011) no. 3, pp. 555-580
10.1353/ajm.2011.0022
[20]
[20] Hacking, Paul; Prokhorov, Yuri Smoothable del Pezzo surfaces with quotient singularities, Compos. Math., Volume 146 (2010) no. 1, pp. 169-192
10.1112/s0010437x09004370
[21]
[21] Hardy, Godfrey H.; Wright, Edward M. An introduction to the theory of numbers, Oxford University Press, 2008 (revised by D. R. Heath-Brown and J. H. Silverman, with a foreword by Andrew Wiles)
10.1093/oso/9780199219858.001.0001
[22]
[22] Hom, Jennifer Bordered Heegaard Floer homology and the tau-invariant of cable knots, J. Topol., Volume 7 (2014) no. 2, pp. 287-326
10.1112/jtopol/jtt030
[23]
[23] Hom, Jennifer; Wu, Zhongtao Four–ball genus bounds and a refinement of the Ozsvath–Szabo tau–invariant, J. Symplectic Geom., Volume 14 (2016) no. 1, pp. 305-323
10.4310/jsg.2016.v14.n1.a12
[24]
[24] Hurwitz, Adolf Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche, Math. Ann., Volume 39 (1891) no. 2, pp. 279-284
10.1007/bf01206656
[25]
[25] Kirby, Robion C. Problems in low-dimensional topology, Geometric topology (Athens, GA, 1993 (Kirby, Robion C., ed.) (AMS/IP Studies in Advanced Mathematics), Volume 2, American Mathematical Society (1997), pp. 35-473
[26]
[26] Larson, Kyle Lattices and correction terms, Singularities and their interaction with geometry and low dimensional topology. In honor of András Némethi on the occasion of his 60th birthday. Selected papers based on the presentations at the conference “Némethi60: geometry and topology of singularities”, Budapest, Hungary, May 27–31, 2019 (Trends in Mathematics), Birkhäuser (2021), pp. 247-257
10.1007/978-3-030-61958-9_11
[27]
[27] Lecuona, Ana G. On the slice-ribbon conjecture for Montesinos knots, Trans. Am. Math. Soc., Volume 364 (2012) no. 1, pp. 233-285
10.1090/s0002-9947-2011-05385-7
[28]
[28] Lecuona, Ana G. On the slice-ribbon conjecture for pretzel knots, Algebr. Geom. Topol., Volume 15 (2015) no. 4, pp. 2133-2173
10.2140/agt.2015.15.2133
[29]
[29] Lecuona, Ana G.; Lisca, Paolo Stein fillable Seifert fibered 3-manifolds, Algebr. Geom. Topol., Volume 11 (2011) no. 2, pp. 625-642
10.2140/agt.2011.11.625
[30]
[30] Lisca, Paolo Lens spaces, rational balls and the ribbon conjecture, Geom. Topol., Volume 11 (2007), pp. 429-472
10.2140/gt.2007.11.429
[31]
[31] Lisca, Paolo Sums of lens spaces bounding rational balls, Algebr. Geom. Topol., Volume 7 (2007), pp. 2141-2164
10.2140/agt.2007.7.2141
[32]
[32] Markoff, Andreĭ A. Sur les formes quadratiques binaires indéfinies, Math. Ann., Volume 17 (1880) no. 3, pp. 379-399
10.1007/bf01446234
[33]
[33] Miller, Allison N. The topological sliceness of 3-strand pretzel knots, Algebr. Geom. Topol., Volume 17 (2017) no. 5, pp. 3057-3079
10.2140/agt.2017.17.3057
[34]
[34] Montesinos, José M. Seifert manifolds that are ramified two-sheeted cyclic coverings, Bol. Soc. Mat. Mex., II. Ser., Volume 18 (1973), pp. 1-32
[35]
[35] Moser, Louise E. Elementary surgery along a torus knot, Pac. J. Math., Volume 38 (1971) no. 3, pp. 737-745
10.2140/pjm.1971.38.737
[36]
[36] Neumann, Walter D. A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Am. Math. Soc., Volume 268 (1981) no. 2, pp. 299-344
10.2307/1999331
[37]
[37] Neumann, Walter D.; Raymond, Frank Seifert manifolds, plumbing, μ-invariant and orientation reversing maps, Algebraic and geometric topology (Proc. Sympos., Univ. California, Santa Barbara, Calif., 1977) (Lecture Notes in Mathematics), Volume 664, Springer, 1978, pp. 163-196
[38]
[38] Ni, Yi; Wu, Zhongtao Cosmetic surgeries on knots in S 3 , J. Reine Angew. Math., Volume 706 (2015), pp. 1-17
10.1515/crelle-2013-0067
[39]
[39] Owens, Brendan Equivariant embeddings of rational homology balls, Q. J. Math., Volume 69 (2018) no. 3, pp. 1101-1121
10.1093/qmath/hay016
[40]
[40] Owens, Brendan; Strle, Sašo Dehn surgeries and negative-definite four-manifolds, Sel. Math., New Ser., Volume 18 (2012) no. 4, pp. 839-854
10.1007/s00029-012-0086-2
[41]
[41] Ozsváth, Peter S.; Szabó, Zoltán Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math., Volume 173 (2003) no. 2, pp. 179-261
10.1016/s0001-8708(02)00030-0
[42]
[42] Ozsváth, Peter S.; Szabó, Zoltán Knot Floer homology and the four-ball genus, Geom. Topol., Volume 7 (2003), pp. 615-639
10.2140/gt.2003.7.615
[43]
[43] Ozsváth, Peter S.; Szabó, Zoltán On the Floer homology of plumbed three-manifolds, Geom. Topol., Volume 7 (2003), pp. 185-224
10.2140/gt.2003.7.185
[44]
[44] Ozsváth, Peter S.; Szabó, Zoltán Knot Floer homology and integer surgeries, Algebr. Geom. Topol., Volume 8 (2008) no. 1, pp. 101-153
10.2140/agt.2008.8.101
[45]
[45] Rasmussen, Jacob Lens space surgeries and a conjecture of Goda and Teragaito, Geom. Topol., Volume 8 (2004), pp. 1013-1031
10.2140/gt.2004.8.1013
[46]
[46] Riemenschneider, Oswald Deformationen von Quotientensingularitaten (nach zyklischen Gruppen), Math. Ann., Volume 209 (1974), pp. 211-248
10.1007/bf01351850
[47]
[47] Shivakumar, Pappur N.; Chew, Kim Ho A Sufficient Condition for Nonvanishing of Determinants, Proc. Am. Math. Soc., Volume 43 (1974) no. 1, pp. 63-66
10.2307/2039326
[48]
[48] Stipsicz, András I.; Szabó, Zoltán; Wahl, Jonathan Rational blowdowns and smoothing of surface singularities, J. Topol., Volume 1 (2008) no. 2, pp. 447-517
10.1112/jtopol/jtn009
[49]
[49] Swenton, Frank Kirby calculator (http://www.klo-software.net/)
[50]
[50] Wu, Zhongtao A cabling formula for the ν + invariant, Proc. Am. Math. Soc., Volume 144 (2016) no. 9, pp. 4089-4098
10.1090/proc/13029
Metrics
0
Citations
50
References
Details
- Published
- Mar 26, 2026
- Pages
- 1-132
Authors
Cite This Article
Paolo Aceto, Marco Golla, Kyle Larson, et al. (2026). Surgeries on torus knots, rational balls, and cabling. Annales de l'Institut Fourier, 1-132. https://doi.org/10.5802/aif.3772
Related