Dehn fillings of knot manifolds containing essential once-punctured tori

Boyer, Steven; Gordon, Cameron; Zhang, Xingru

doi:10.1090/S0002-9947-2013-05837-0

Dehn fillings of knot manifolds containing essential once-punctured tori
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by Steven Boyer, Cameron McA. Gordon and Xingru Zhang PDF

Trans. Amer. Math. Soc. 366 (2014), 341-393 Request permission

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 $|n|>1$. 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

Kenneth L. Baker, Once-punctured tori and knots in lens spaces, Comm. Anal. Geom. 19 (2011), no. 2, 347–399. MR 2835883, DOI 10.4310/CAG.2011.v19.n2.a4
Joan S. Birman and William W. Menasco, Studying links via closed braids. III. Classifying links which are closed $3$-braids, Pacific J. Math. 161 (1993), no. 1, 25–113. MR 1237139, DOI 10.2140/pjm.1993.161.25
Steve Boyer, Marc Culler, Peter B. Shalen, and Xingru Zhang, Characteristic subsurfaces and Dehn filling, Trans. Amer. Math. Soc. 357 (2005), no. 6, 2389–2444. MR 2140444, DOI 10.1090/S0002-9947-04-03576-7
Steve Boyer, Marc Culler, Peter B. Shalen, and Xingru Zhang, Characteristic subsurfaces, character varieties and Dehn fillings, Geom. Topol. 12 (2008), no. 1, 233–297. MR 2390346, DOI 10.2140/gt.2008.12.233
S. Boyer, C. McA. Gordon, and X. Zhang, Dehn fillings of large hyperbolic 3-manifolds, J. Differential Geom. 58 (2001), no. 2, 263–308. MR 1913944, DOI 10.4310/jdg/1090348327
—, Characteristic submanifold theory and toroidal Dehn filling, Adv. Math. 230 (2012), 1673–1737.
—, Dehn fillings of knot manifolds containing essential twice-punctured tori, in preparation.
S. Boyer and X. Zhang, Reducing Dehn filling and toroidal Dehn filling, Topology Appl. 68 (1996), no. 3, 285–303. MR 1377050, DOI 10.1016/0166-8641(95)00061-5
S. Boyer and X. Zhang, On Culler-Shalen seminorms and Dehn filling, Ann. of Math. (2) 148 (1998), no. 3, 737–801. MR 1670053, DOI 10.2307/121031
Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987), no. 2, 237–300. MR 881270, DOI 10.2307/1971311
M. Culler, W. Jaco, and H. Rubinstein, Incompressible surfaces in once-punctured torus bundles, Proc. London Math. Soc. (3) 45 (1982), no. 3, 385–419. MR 675414, DOI 10.1112/plms/s3-45.3.385
William D. Dunbar, Geometric orbifolds, Rev. Mat. Univ. Complut. Madrid 1 (1988), no. 1-3, 67–99. MR 977042
Mario Eudave-Muñoz, Non-hyperbolic manifolds obtained by Dehn surgery on hyperbolic knots, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 35–61. MR 1470720, DOI 10.1090/amsip/002.1/03
W. Floyd and A. Hatcher, Incompressible surfaces in punctured-torus bundles, Topology Appl. 13 (1982), no. 3, 263–282. MR 651509, DOI 10.1016/0166-8641(82)90035-9
David Futer, Efstratia Kalfagianni, and Jessica S. Purcell, Cusp areas of Farey manifolds and applications to knot theory, Int. Math. Res. Not. IMRN 23 (2010), 4434–4497. MR 2739802, DOI 10.1093/imrn/rnq037
David Gabai, Foliations and the topology of $3$-manifolds. II, J. Differential Geom. 26 (1987), no. 3, 461–478. MR 910017
C. McA. Gordon, Boundary slopes of punctured tori in $3$-manifolds, Trans. Amer. Math. Soc. 350 (1998), no. 5, 1713–1790. MR 1390037, DOI 10.1090/S0002-9947-98-01763-2
C. McA. Gordon, Dehn filling: a survey, Knot theory (Warsaw, 1995) Banach Center Publ., vol. 42, Polish Acad. Sci. Inst. Math., Warsaw, 1998, pp. 129–144. MR 1634453
C. McA. Gordon and R. A. Litherland, Incompressible surfaces in branched coverings, The Smith conjecture (New York, 1979) Pure Appl. Math., vol. 112, Academic Press, Orlando, FL, 1984, pp. 139–152. MR 758466, DOI 10.1016/S0079-8169(08)61639-6
C. McA. Gordon and J. Luecke, Toroidal and boundary-reducing Dehn fillings, Topology Appl. 93 (1999), no. 1, 77–90. MR 1684214, DOI 10.1016/S0166-8641(97)00258-7
Cameron McA. Gordon and Ying-Qing Wu, Toroidal Dehn fillings on hyperbolic 3-manifolds, Mem. Amer. Math. Soc. 194 (2008), no. 909, vi+140. MR 2419168, DOI 10.1090/memo/0909
Cameron McA. Gordon and Ying-Qing Wu, Toroidal and annular Dehn fillings, Proc. London Math. Soc. (3) 78 (1999), no. 3, 662–700. MR 1674841, DOI 10.1112/S0024611599001823
Craig Hodgson and J. H. Rubinstein, Involutions and isotopies of lens spaces, Knot theory and manifolds (Vancouver, B.C., 1983) Lecture Notes in Math., vol. 1144, Springer, Berlin, 1985, pp. 60–96. MR 823282, DOI 10.1007/BFb0075012
Paik Kee Kim and Jeffrey L. Tollefson, Splitting the PL involutions of nonprime $3$-manifolds, Michigan Math. J. 27 (1980), no. 3, 259–274. MR 584691
M. Lackenby and R. Meyerhoff, The maximal number of exceptional Dehn surgeries, preprint (2008), arXiv:0808.1176.
Sangyop Lee, Dehn fillings yielding Klein bottles, Int. Math. Res. Not. , posted on (2006), Art. ID 24253, 34. MR 2219228, DOI 10.1155/IMRN/2006/24253
Sangyop Lee, Klein bottle and toroidal Dehn fillings at distance 5, Pacific J. Math. 247 (2010), no. 2, 407–434. MR 2734156, DOI 10.2140/pjm.2010.247.407
Sangyop Lee, Lens spaces and toroidal Dehn fillings, Math. Z. 267 (2011), no. 3-4, 781–802. MR 2776058, DOI 10.1007/s00209-009-0646-0
Sangyop Lee and Masakazu Teragaito, Boundary structure of hyperbolic 3-manifolds admitting annular and toroidal fillings at large distance, Canad. J. Math. 60 (2008), no. 1, 164–188. MR 2381171, DOI 10.4153/CJM-2008-007-6
Bruno Martelli and Carlo Petronio, Dehn filling of the “magic” 3-manifold, Comm. Anal. Geom. 14 (2006), no. 5, 969–1026. MR 2287152, DOI 10.4310/CAG.2006.v14.n5.a6
Yoav Moriah and Jennifer Schultens, Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal, Topology 37 (1998), no. 5, 1089–1112. MR 1650355, DOI 10.1016/S0040-9383(97)00072-4
Yi Ni, Knot Floer homology detects fibred knots, Invent. Math. 170 (2007), no. 3, 577–608. MR 2357503, DOI 10.1007/s00222-007-0075-9
Ulrich Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984), no. 1, 209–230. MR 732067, DOI 10.2140/pjm.1984.111.209
Seungsang Oh, Reducible and toroidal $3$-manifolds obtained by Dehn fillings, Topology Appl. 75 (1997), no. 1, 93–104. MR 1425387, DOI 10.1016/S0166-8641(96)00065-X
Hamish Short, Some closed incompressible surfaces in knot complements which survive surgery, Low-dimensional topology (Chelwood Gate, 1982) London Math. Soc. Lecture Note Ser., vol. 95, Cambridge Univ. Press, Cambridge, 1985, pp. 179–194. MR 827302, DOI 10.1017/CBO9780511662744.007
Masakazu Teragaito, Distance between toroidal surgeries on hyperbolic knots in the 3-sphere, Trans. Amer. Math. Soc. 358 (2006), no. 3, 1051–1075. MR 2187645, DOI 10.1090/S0002-9947-05-03703-7
Masakazu Teragaito, Creating Klein bottles by surgery on knots, J. Knot Theory Ramifications 10 (2001), no. 5, 781–794. Knots in Hellas ’98, Vol. 3 (Delphi). MR 1839702, DOI 10.1142/S0218216501001153
Ying Qing Wu, Incompressibility of surfaces in surgered $3$-manifolds, Topology 31 (1992), no. 2, 271–279. MR 1167169, DOI 10.1016/0040-9383(92)90020-I
Ying-Qing Wu, Dehn fillings producing reducible manifolds and toroidal manifolds, Topology 37 (1998), no. 1, 95–108. MR 1480879, DOI 10.1016/S0040-9383(97)00016-5
Shing-Tung Yau and William H. Meeks III, The equivariant loop theorem for three-dimensional manifolds and a review of the existence theorems for minimal surfaces, The Smith conjecture (New York, 1979) Pure Appl. Math., vol. 112, Academic Press, Orlando, FL, 1984, pp. 153–163. MR 758467, DOI 10.1016/S0079-8169(08)61640-2