Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Dehn fillings of knot manifolds containing essential once-punctured tori


Authors: Steven Boyer, Cameron McA. Gordon and Xingru Zhang
Journal: Trans. Amer. Math. Soc. 366 (2014), 341-393
MSC (2010): Primary 57M25, 57M50, 57M99
DOI: https://doi.org/10.1090/S0002-9947-2013-05837-0
Published electronically: June 10, 2013
MathSciNet review: 3118399
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study exceptional Dehn fillings on hyperbolic knot manifolds which contain an essential once-punctured torus. Let $ M$ be such a knot manifold and let $ \beta $ be the boundary slope of such an essential once-punctured torus. We prove that if Dehn filling $ M$ with slope $ \alpha $ produces a Seifert fibred manifold, then $ \Delta (\alpha ,\beta )\leq 5$. Furthermore we classify the triples $ (M; \alpha ,\beta )$ when $ \Delta (\alpha ,\beta )\geq 4$. More precisely, when $ \Delta (\alpha ,\beta )=5$, then $ M$ is the (unique) manifold $ Wh(-3/2)$ obtained by Dehn filling one boundary component of the Whitehead link exterior with slope $ -3/2$, and $ (\alpha , \beta )$ is the pair of slopes $ (-5, 0)$. Further, $ \Delta (\alpha ,\beta )=4$ if and only if $ (M; \alpha ,\beta )$ is the triple $ \displaystyle (Wh(\frac {-2n\pm 1}{n}); -4, 0)$ for some integer $ n$ with $ \vert n\vert>1$. Combining this with known results, we classify all hyperbolic knot manifolds $ M$ and pairs of slopes $ (\beta , \gamma )$ on $ \partial M$ where $ \beta $ is the boundary slope of an essential once-punctured torus in $ M$ and $ \gamma $ is an exceptional filling slope of distance $ 4$ or more from $ \beta $. Refined results in the special case of hyperbolic genus one knot exteriors in $ S^3$ are also given.


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

  • [Ba] K. Baker, Once-punctured tori and knots in lens spaces, Comm. Anal. Geom. 19 (2011), 347-399. MR 2835883
  • [BiMe] J. Birman and Wm. Menasco, Studying links via closed braids III, Pacific J. Math. 161 (1993), 25-113. MR 1237139 (94i:57005)
  • [BCSZ1] S. Boyer, M. Culler, P. Shalen, and X. Zhang, Characteristic submanifold theory and Dehn filling, Trans. Amer. Math. Soc. 357 (2005), 2389-2444. MR 2140444 (2006a:57018)
  • [BCSZ2] -, Characteristic subvarieties, character varieties, and Dehn fillings, Geometry & Topology 12 (2008) 233-297. MR 2390346 (2009a:57003)
  • [BGZ1] S. Boyer, C. McA. Gordon and X. Zhang, Dehn fillings of large hyperbolic 3-manifolds, J. Diff. Geom. 58 (2001), 263-308. MR 1913944 (2003j:57025)
  • [BGZ2] -, Characteristic submanifold theory and toroidal Dehn filling, Adv. Math. 230 (2012), 1673-1737.
  • [BGZ3] -, Dehn fillings of knot manifolds containing essential twice-punctured tori, in preparation.
  • [BZ1] S. Boyer and X. Zhang, Reducing Dehn filling and toroidal Dehn filling, Topology Appl. 68 (1996) 285-303. MR 1377050 (97f:57018)
  • [BZ2] -, On Culler-Shalen seminorms and Dehn fillings, Ann. Math. 148 (1998), 737-801. MR 1670053 (2000d:57028)
  • [CGLS] M. Culler, C. M. Gordon, J. Luecke and P. Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987) 237-300. MR 881270 (88a:57026)
  • [CJR] M. Culler, W. Jaco and H. Rubinstein, Incompressible surfaces in once-punctured torus bundles, Proc. Lond. Math. Soc. 45 (1982) 385-419. MR 675414 (84a:57010)
  • [Du] Wm. Dunbar, Geometric orbifolds, Rev. Mat. 1 (1988) 67-99. MR 977042 (90k:22011)
  • [E-M] M. Eudave-Muñoz, Non-hyperbolic manifolds obtained by Dehn surgery on hyperbolic knots, Geometric topology (Athens, GA, 1993), 35-61, AMS/IP Stud. Adv. Math., 2.1, Amer. Math. Soc., Providence, RI, 1997. MR 1470720 (98i:57007)
  • [FH] W. Floyd and A. Hatcher, Incompressible surfaces in punctured-torus bundles, Top. Appl. 13 (1982), 263-282. MR 651509 (83h:57015)
  • [FKP] D. Futer, E. Kalfagianni, and J. Purcell, Cusp areas of Farey manifolds and applications to knot theory, Int. Math. Res. Not. 2010, no. 23, 4434-4497. MR 2739802 (2011k:57027)
  • [Ga] D. Gabai, Foliations and the topology of $ 3$-manifolds II, J. Diff. Geom. 26 (1987), 461-478. MR 910017 (89a:57014a)
  • [Go1] C. McA. Gordon, Boundary slopes of punctured tori in $ 3$-manifolds, Trans. Amer. Math. Soc. 350 (1998) 1713-1790. MR 1390037 (98h:57032)
  • [Go2] -, Dehn filling: a survey, in Knot Theory, Banach Center Publications 42, Institute of Mathematics, Polish Academy of Sciences, Warsaw, 1998, 129-144. MR 1634453 (99e:57028)
  • [GLi] C. McA. Gordon and R. A. Litherland, Incompressible surfaces in branched coverings, The Smith conjecture, Pure Appl. Math., 112, Academic Press (1984) 139-152. MR 758466
  • [GL] C. McA. Gordon and J. Luecke, Toroidal and boundary-reducing Dehn Fillings, Topology Appl. 93 (1999) 77-90. MR 1684214 (2000b:57030)
  • [GW] C. McA. Gordon and Y.-Q. Wu, Toroidal Dehn fillings on hyperbolic 3-manifolds, Mem. Amer. Math. Soc. 194 (2008), no.909. MR 2419168 (2009c:57036)
  • [GW2] -, Toroidal and annular Dehn fillings, Proc. London Math. Soc. 78 (1999) 662-700. MR 1674841 (2000b:57029)
  • [HR] C. Hodgson and H. Rubinstein, Involutions and isotopies of lens spaces, in Knot Theory and Manifolds, ed. D. Rolfsen, Lecture Notes in Mathematics 1144, Springer-Verlag, 1983, 60-96. MR 823282 (87h:57028)
  • [KT] P. K. Kim and J. L. Tollefson, Splitting the PL involutions of nonprime $ 3$-manifolds, Michigan Math. J. 27 (1980) 259-274. MR 584691 (81m:57007)
  • [LM] M. Lackenby and R. Meyerhoff, The maximal number of exceptional Dehn surgeries, preprint (2008), arXiv:0808.1176.
  • [L1] S. Lee, Dehn fillings yielding Klein bottles, Int. Math. Res. Not. 2006, Art. ID 24253, 34pp. MR 2219228 (2007b:57038)
  • [L2] -, Klein bottle and toroidal Dehn fillings at distance 5, Pacific J. Math. 247 (2010), 407-434. MR 2734156 (2012e:57036)
  • [L3] -, Lens spaces and toroidal Dehn fillings, Math. Z. 267 (2011), 781-802. MR 2776058 (2012a:57022)
  • [LT] S. Lee and M. Teragaito, Boundary structure of hyperbolic 3-manifolds admitting annular and toroidal fillings at large distance, Canad. J. Math. 60 (2008) 164-188. MR 2381171 (2009a:57029)
  • [MP] B. Martelli and C. Petronio, Dehn filling of the ``magic'' 3-manifold, Comm. Anal. Geom. 14 (2006), 969-1026. MR 2287152 (2007k:57042)
  • [MSch] Y. Moriah and J. Schultens, Irreducible Heegaard splittings of Seifert fibred spaces are either vertical or horizontal, Topology 37 (1998) 1089-1112. MR 1650355 (99g:57021)
  • [Ni] Y. Ni, Knot Floer homology detects fibred knots, Invent. Math. 170 (2007), no. 3, 577-608. MR 2357503 (2008j:57053)
  • [Oe] U. Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984), 209-230. MR 732067 (85j:57008)
  • [Oh] S. Oh, Reducible and toroidal manifolds obtained by Dehn filling, Top. Appl. 75 (1997), 93-104. MR 1425387 (98a:57027)
  • [Sh] H. Short, Some closed incompressible surfaces in knot complements which survive surgery, in Low dimensional topology, ed. Roger Fenn, London Math. Soc. Lecture Notes 95, Cambridge University Press, 1985, 179-194. MR 827302 (88d:57006)
  • [Te1] M. Teragaito, Distance between toroidal surgeries on hyperbolic knots in the 3-sphere, Trans. Amer. Math. Soc. 358 (2006), 1051-1075. MR 2187645 (2006h:57005)
  • [Te2] -, Creating Klein bottles by surgery on knots, J. Knot Theory Ramifications 10 (2001) 781-794. MR 1839702 (2002f:57017)
  • [Wu1] Y.-Q. Wu, Incompressibility of surfaces in surgered 3-manifolds, Topology, 31 (1992) 271-279. MR 1167169 (94e:57027)
  • [Wu2] -, Dehn fillings producing reducible manifolds and toroidal manifolds, Topology 37 (1998), 95-108. MR 1480879 (98j:57033)
  • [YM] S-T Yau and W. Meeks, The equivariant loop theorem for three-dimensional manifolds and a review of the existence theorem for minimal surfaces, in The Smith Conjecture, Pure Appl. Math., 112, Academic Press (1984) 153-163. MR 758467

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 57M25, 57M50, 57M99

Retrieve articles in all journals with MSC (2010): 57M25, 57M50, 57M99


Additional Information

Steven Boyer
Affiliation: Département de Mathématiques, Université du Québec à Montréal, 201 avenue du Président-Kennedy, Montréal, Québec, Canada H2X 3Y7
Email: boyer.steven@uqam.ca

Cameron McA. Gordon
Affiliation: Department of Mathematics, University of Texas at Austin, 1 University Station, Austin, Texas 78712
Email: gordon@math.utexas.edu

Xingru Zhang
Affiliation: Department of Mathematics, SUNY at Buffalo, Buffalo, New York 14214-3093
Email: xinzhang@buffalo.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-05837-0
Received by editor(s): November 8, 2011
Received by editor(s) in revised form: March 23, 2012
Published electronically: June 10, 2013
Additional Notes: The first author was partially supported by NSERC grant RGPIN 9446-2008
The second author was partially supported by NSF grant DMS-0906276.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society