The lens space realization problem. (English) Zbl 1276.57009

A knot \(K\) in \(S^3\) is a lens space knot if it admits a lens space surgery. There is a question asked by Moser: What are all the ways to construct lens spaces \(L(p, q)\) by Dehn surgeries along knots in \(S^3\)? The answer to this question remains open.
One of the best results was due to Berge, who enumerated many lens space knots, which are doubly primitive knots. The doubly primitive knots that Berge enumerated are called Berge knots. Berge further conjectured that his construction yields all the knots that allow integer lens space surgeries. On the other hand, Culler-Gordon-Luecke-Shalen proved that if a lens space knot is not a torus knot, then the surgery coefficient is an integer. So, to answer Moser’s question we just need to prove Berge’s conjecture affirmatively.
There are three questions that are closely related to this conjecture.
Q1: How to realize the lens spaces that are obtained from integer surgery along a knot in \(S^3\)?
Q2: Do the Berge knots include all the doubly primitive knots?
Q3: Can we bound the knot genus \(g(K)\) from above tightly in terms of the surgery slope \(p\)?
The following results in the paper under review answer these three questions.
{Theorem 1.2.} Suppose that negative integer surgery along a knot \(K\subset L(p,q)\) produces \(S^3\). Then \(K\) lies in the same homology class as a Berge knot \(B \subset L(p, q)\).
{Theorem 1.3.} Suppose that \(K \subset S^3\), \(p\) is a positive integer, and the surgered manifold by \(p\)-surgery along \(K\), \(K_p\), is a lens space. Then there exists a Berge knot \(B\subset S^3\) such that \(B_p \cong K_p\) and \(\widehat{HFK}(B)\cong\widehat{HFK}(K)\). Furthermore, every doubly primitive knot in \(S^3\) is a Berge knot.
{Theorem 1.4.} Suppose that \(K \subset S^3\), \(p\) is a positive integer, and \(K_p\) is a lens space. Then \(2g(K)-1\leq p-2 \sqrt{(4p+1)/5}\), unless \(K\) is the right-hand trefoil and \(p = 5\). Moreover, this bound is attained by the type VIII Berge knots specified by the pairs \((p, k) = (5n^2 +5n+1, 5n^2-1)\).
Reviewer: Yu Zhang (Harbin)


57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)


Full Text: DOI arXiv


[1] J. Bailey and D. Rolfsen, ”An unexpected surgery construction of a lens space,” Pacific J. Math., vol. 71, iss. 2, pp. 295-298, 1977. · Zbl 0349.55004
[2] K. L. Baker, The Poincaré homology sphere and almost-simple knots in lens spaces, 2011. · Zbl 1287.57016
[3] K. L. Baker, Private communication, 2011.
[4] K. L. Baker, E. J. Grigsby, and M. Hedden, ”Grid diagrams for lens spaces and combinatorial knot Floer homology,” Int. Math. Res. Not., vol. 2008, iss. 10, p. I. · Zbl 1168.57009
[5] J. Berge, ”The knots in \(D^2\times S^1\) which have nontrivial Dehn surgeries that yield \(D^2\times S^1\),” Topology Appl., vol. 38, iss. 1, pp. 1-19, 1991. · Zbl 0725.57001
[6] J. Berge, The simple closed curves in genus two Heegaard surfaces of \(S^3\) which are double-primitives, 2010.
[7] J. Berge, Some knots with surgeries yielding lens spaces, c. 1990.
[8] J. L. Brown Jr., ”Note on complete sequences of integers,” Amer. Math. Monthly, vol. 68, pp. 557-560, 1961. · Zbl 0115.04305
[9] A. J. Casson and . C. Gordon, ”Cobordism of classical knots,” in À la recherche de la Topologie Perdue, Boston, MA: Birkhäuser, 1986, vol. 62, pp. 181-199. · Zbl 0597.57001
[10] M. Culler, . C. Gordon, J. Luecke, and P. B. Shalen, ”Dehn surgery on knots,” Ann. of Math., vol. 125, iss. 2, pp. 237-300, 1987. · Zbl 0633.57006
[11] A. Donald and B. Owens, Concordance groups of links, 2012. · Zbl 1266.57005
[12] S. K. Donaldson, ”The orientation of Yang-Mills moduli spaces and \(4\)-manifold topology,” J. Differential Geom., vol. 26, iss. 3, pp. 397-428, 1987. · Zbl 0683.57005
[13] N. D. Elkies, ”A characterization of the \({\mathbf Z}^n\) lattice,” Math. Res. Lett., vol. 2, iss. 3, pp. 321-326, 1995. · Zbl 0855.11032
[14] R. Fintushel and R. J. Stern, Private communication, 2012.
[15] R. Fintushel and R. J. Stern, ”Constructing lens spaces by surgery on knots,” Math. Z., vol. 175, iss. 1, pp. 33-51, 1980. · Zbl 0425.57001
[16] R. Fintushel and R. J. Stern, ”A \(\mu\)-invariant one homology \(3\)-sphere that bounds an orientable rational ball,” in Four-Manifold Theory, Providence, RI: Amer. Math. Soc., 1984, vol. 35, pp. 265-268. · Zbl 0566.57006
[17] K. A. Frøyshov, ”The Seiberg-Witten equations and four-manifolds with boundary,” Math. Res. Lett., vol. 3, iss. 3, pp. 373-390, 1996. · Zbl 0872.57024
[18] D. Gabai, ”Surgery on knots in solid tori,” Topology, vol. 28, iss. 1, pp. 1-6, 1989. · Zbl 0678.57004
[19] L. J. Gerstein, ”Nearly unimodular quadratic forms,” Ann. of Math., vol. 142, iss. 3, pp. 597-610, 1995. · Zbl 0842.11012
[20] P. Ghiggini, ”Knot Floer homology detects genus-one fibred knots,” Amer. J. Math., vol. 130, iss. 5, pp. 1151-1169, 2008. · Zbl 1149.57019
[21] H. Goda and M. Teragaito, ”Dehn surgeries on knots which yield lens spaces and genera of knots,” Math. Proc. Cambridge Philos. Soc., vol. 129, iss. 3, pp. 501-515, 2000. · Zbl 1029.57005
[22] J. Greene, On closed 3-braids with unknotting number one, 2009.
[23] J. Greene, L-space surgeries, genus bounds, and the cabling conjecture, 2010. · Zbl 1325.57016
[24] J. Greene, Lattices, graphs, and Conway mutation, 2011. · Zbl 1278.57021
[25] J. Greene and S. Jabuka, ”The slice-ribbon conjecture for 3-stranded pretzel knots,” Amer. J. Math., vol. 133, iss. 3, pp. 555-580, 2011. · Zbl 1225.57006
[26] M. Hedden, ”On Floer homology and the Berge conjecture on knots admitting lens space surgeries,” Trans. Amer. Math. Soc., vol. 363, iss. 2, pp. 949-968, 2011. · Zbl 1229.57006
[27] M. A. Kervaire and J. W. Milnor, ”On \(2\)-spheres in \(4\)-manifolds,” Proc. Nat. Acad. Sci. U.S.A., vol. 47, pp. 1651-1657, 1961. · Zbl 0107.40303
[28] R. Kirby, Problems in low-dimensional topology, 2010. · Zbl 0394.57002
[29] P. Kronheimer, T. Mrowka, P. Ozsváth, and Z. Szabó, ”Monopoles and lens space surgeries,” Ann. of Math., vol. 165, iss. 2, pp. 457-546, 2007. · Zbl 1204.57038
[30] A. G. Lecuona, ”On the slice-ribbon conjecture for Montesinos knots,” Trans. Amer. Math. Soc., vol. 364, iss. 1, pp. 233-285, 2012. · Zbl 1244.57017
[31] P. Lisca, ”Lens spaces, rational balls and the ribbon conjecture,” Geom. Topol., vol. 11, pp. 429-472, 2007. · Zbl 1185.57006
[32] P. Lisca, ”Sums of lens spaces bounding rational balls,” Algebr. Geom. Topol., vol. 7, pp. 2141-2164, 2007. · Zbl 1185.57015
[33] D. McDuff and F. Schlenk, ”The embedding capacity of 4-dimensional symplectic ellipsoids,” Ann. of Math., vol. 175, iss. 3, pp. 1191-1282, 2012. · Zbl 1254.53111
[34] J. Milnor and D. Husemoller, Symmetric Bilinear Forms, New York: Springer-Verlag, 1973, vol. 73. · Zbl 0292.10016
[35] L. Moser, ”Elementary surgery along a torus knot,” Pacific J. Math., vol. 38, pp. 737-745, 1971. · Zbl 0202.54701
[36] Y. Ni, ”Knot Floer homology detects fibred knots,” Invent. Math., vol. 170, iss. 3, pp. 577-608, 2007. · Zbl 1138.57031
[37] Y. Ni, ”Erratum: Knot Floer homology detects fibred knots,” Invent. Math., vol. 177, iss. 1, pp. 235-238, 2009. · Zbl 1162.57300
[38] P. Ozsváth and Z. Szabó, ”Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary,” Adv. Math., vol. 173, iss. 2, pp. 179-261, 2003. · Zbl 1025.57016
[39] P. Ozsváth and Z. Szabó, ”On knot Floer homology and lens space surgeries,” Topology, vol. 44, iss. 6, pp. 1281-1300, 2005. · Zbl 1077.57012
[40] J. Rasmussen, Lens space surgeries and L-space homology spheres, 2007.
[41] S. Rasmussen, A number theoretic result for Berge’s conjecture, 2009.
[42] O. Riemenschneider, ”Deformationen von Quotientensingularitäten (nach zyklischen Gruppen),” Math. Ann., vol. 209, pp. 211-248, 1974. · Zbl 0275.32010
[43] T. Saito, ”A note on lens space surgeries: orders of fundamental groups versus Seifert genera,” J. Knot Theory Ramifications, vol. 20, iss. 4, pp. 617-624, 2011. · Zbl 1222.57005
[44] M. Tange, ”Lens spaces given from L-space homology 3-spheres,” Experiment. Math., vol. 18, iss. 3, pp. 285-301, 2009. · Zbl 1187.57011
[45] M. Tange, A complete list of lens spaces constructed by Dehn surgery I, 2010.
[46] M. Tange, Private communication, 2011.
[47] S. Wolfram, The Mathematica\(^\circledR\) Book, Fourth ed., Wolfram Media, Champaign, IL, 1999. · Zbl 0924.65002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.