Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the number of finite $ p/q$-surgeries

Author: Margaret I. Doig
Journal: Proc. Amer. Math. Soc. 144 (2016), 2205-2215
MSC (2010): Primary 51M09, 57M25, 57R65
Published electronically: October 20, 2015
MathSciNet review: 3460179
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study finite, non-cyclic knot surgeries, that is, surgeries which give manifolds of finite but not cyclic fundamental group. These manifolds are known to be knot surgeries except for the dihedral manifolds. We show that, for a fixed $ p$, there are finitely many dihedral manifolds that are $ p/q$-surgery, and we place a bound on which manifolds they may be. In the process, we calculate a recursive relationship among the Heegaard Floer d-invariants of dihedral manifolds with a given first homology and calculate a bound on which d-invariants would occur if such a manifold were surgery on a knot in $ S^3$.

References [Enhancements On Off] (What's this?)

  • [BH96] Steven A. Bleiler and Craig D. Hodgson, Spherical space forms and Dehn filling, Topology 35 (1996), no. 3, 809-833. MR 1396779 (97f:57007),
  • [BZ96] S. Boyer and X. Zhang, Finite Dehn surgery on knots, J. Amer. Math. Soc. 9 (1996), no. 4, 1005-1050. MR 1333293 (97h:57013),
  • [BZ01] Steven Boyer and Xingru Zhang, A proof of the finite filling conjecture, J. Differential Geom. 59 (2001), no. 1, 87-176. MR 1909249 (2003k:57007)
  • [Doi15] Margaret I. Doig, Finite knot surgeries and Heegaard Floer homology, Algebr. Geom. Topol. 15 (2015), no. 2, 667-690. MR 3342672,
  • [FIK$^+$09] David Futer, Masaharu Ishikawa, Yuichi Kabaya, Thomas W. Mattman, and Koya Shimokawa, Finite surgeries on three-tangle pretzel knots, Algebr. Geom. Topol. 9 (2009), no. 2, 743-771. MR 2496889 (2010a:57008),
  • [FS80] Ronald Fintushel and Ronald J. Stern, Constructing lens spaces by surgery on knots, Math. Z. 175 (1980), no. 1, 33-51. MR 595630 (82i:57009a),
  • [FS87] Ronald Fintushel and Ronald Stern, Rational homology cobordisms of spherical space forms, Topology 26 (1987), no. 3, 385-393. MR 899056 (89d:57051),
  • [Gor83] C. McA. Gordon, Dehn surgery and satellite knots, Trans. Amer. Math. Soc. 275 (1983), no. 2, 687-708. MR 682725 (84d:57003),
  • [Gre13] Joshua Evan Greene, The lens space realization problem, Ann. of Math. (2) 177 (2013), no. 2, 449-511. MR 3010805,
  • [HW] Matthew Hedden and Liam Watson, On the geography and botany of knot Floer homology, arXiv:1404.6913.
  • [LN] Eileen Li and Yi Ni, Half-integral finite surgeries on knots in $ {S}^3$, arXiv:1310.1346.
  • [Mat02] Thomas W. Mattman, Cyclic and finite surgeries on pretzel knots, J. Knot Theory Ramifications 11 (2002), no. 6, 891-902. Knots 2000 Korea, Vol. 3 (Yongpyong). MR 1936241 (2003j:57013),
  • [Mos71] Louise Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), 737-745. MR 0383406 (52 #4287)
  • [MT07] John Morgan and Gang Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs, vol. 3, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007. MR 2334563 (2008d:57020)
  • [Ném05] András Némethi, On the Ozsváth-Szabó invariant of negative definite plumbed 3-manifolds, Geom. Topol. 9 (2005), 991-1042. MR 2140997 (2006c:57011),
  • [Ni07] Yi Ni, Knot Floer homology detects fibred knots, Invent. Math. 170 (2007), no. 3, 577-608. MR 2357503 (2008j:57053),
  • [OS03a] Peter Ozsváth and Zoltán Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003), no. 2, 179-261. MR 1957829 (2003m:57066),
  • [OS03b] Peter Ozsváth and Zoltán Szabó, On the Floer homology of plumbed three-manifolds, Geom. Topol. 7 (2003), 185-224 (electronic). MR 1988284 (2004h:57039),
  • [OS05] Peter Ozsváth and Zoltán Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005), no. 6, 1281-1300. MR 2168576 (2006f:57034),
  • [OS11] Peter S. Ozsváth and Zoltán Szabó, Knot Floer homology and rational surgeries, Algebr. Geom. Topol. 11 (2011), no. 1, 1-68. MR 2764036 (2012h:57056),
  • [Sei33] H. Seifert, Topologie Dreidimensionaler Gefaserter Räume, Acta Math. 60 (1933), no. 1, 147-238 (German). MR 1555366,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 51M09, 57M25, 57R65

Retrieve articles in all journals with MSC (2010): 51M09, 57M25, 57R65

Additional Information

Margaret I. Doig
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244

Received by editor(s): November 4, 2013
Received by editor(s) in revised form: February 3, 2015, June 4, 2015, and June 12, 2015
Published electronically: October 20, 2015
Communicated by: Martin Scharlemann
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society