## On hyperbolic 3-manifolds realizing the maximal distance between toroidal Dehn fillings.(English)Zbl 1082.57011

Let $$M$$ be a hyperbolic $$3$$-manifold with a torus boundary component $$T_0$$, in the sense that $$M$$ with its boundary tori removed admits a complete hyperbolic structure with totally geodesic boundary. For a slope $$\gamma$$ on $$T_0$$, $$\gamma$$-Dehn filling $$M(\gamma)$$ is hyperbolic except finitely many slopes, which are referred to as exceptional slopes. The case where $$M(\gamma)$$ contains an essential torus is typical, and such a slope is said to be toroidal. The paper under review focuses on this situation. For two slopes, the distance between them is the minimal geometric intersection number. Roughly speaking, an upper bound for the distance between exceptional slopes gives that for the number of exceptional slopes.
C. Gordon [Trans. Am. Math. Soc. 350, 1713–1790 (1998; Zbl 0896.57011)] showed that if two slopes $$\gamma_1$$ and $$\gamma_2$$ are toroidal then the distance $$\Delta$$ between them is at most eight, and there are only four specific manifolds with $$\Delta\geq 6$$. A manifold $$M$$ is said to be large if $$H_2(M,\partial M-T_0)\not=0$$, following Y. Q. Wu [J. Differ. Geom. 48, 407–437 (1998; Zbl 0917.57015)]. Since the above four specific manifolds are not large, $$\Delta\leq 5$$ for a large $$M$$ manifold. C. McA. Gordon [Geom. Topol. Monogr. 2, 177–199 (1999; Zbl 0948.57014)] asked if there is a large $$M$$ with $$\Delta=5$$.
The main result of the paper under review is that $$\Delta\leq 4$$, unless the boundary of $$M$$ consists of at most two tori. Thus a broad class of large manifolds cannot admit two toroidal slopes at distance five. This is also the first step to determine all hyperbolic $$3$$-manifolds that admit two toroidal slopes at distance five. The proof is based on a long analysis of the graph pair coming from the intersection of two punctured tori as in earlier works in this area.

### MSC:

 57M50 General geometric structures on low-dimensional manifolds 57M25 Knots and links in the $$3$$-sphere (MSC2010)

### Keywords:

Dehn filling; toroidal filling; knot

### Citations:

Zbl 0896.57011; Zbl 0917.57015; Zbl 0948.57014
Full Text:

### References:

 [1] S Boyer, C M Gordon, X Zhang, Dehn fillings of large hyperbolic 3-manifolds, J. Differential Geom. 58 (2001) 263 · Zbl 1042.57007 [2] S Boyer, X Zhang, Reducing Dehn filling and toroidal Dehn filling, Topology Appl. 68 (1996) 285 · Zbl 0852.57002 [3] M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. $$(2)$$ 125 (1987) 237 · Zbl 0633.57006 [4] C M Gordon, Dehn filling: a survey, Banach Center Publ. 42, Polish Acad. Sci. (1998) 129 · Zbl 0916.57016 [5] C M Gordon, Boundary slopes of punctured tori in 3-manifolds, Trans. Amer. Math. Soc. 350 (1998) 1713 · Zbl 0896.57011 [6] C M Gordon, Small surfaces and Dehn filling, Geom. Topol. Monogr. 2, Geom. Topol. Publ., Coventry (1999) 177 · Zbl 0948.57014 [7] C M Gordon, J Luecke, Dehn surgeries on knots creating essential tori I, Comm. Anal. Geom. 3 (1995) 597 · Zbl 0865.57015 [8] C M Gordon, J Luecke, Toroidal and boundary-reducing Dehn fillings, Topology Appl. 93 (1999) 77 · Zbl 0926.57019 [9] C M Gordon, J Luecke, Dehn surgeries on knots creating essential tori II, Comm. Anal. Geom. 8 (2000) 671 · Zbl 0970.57010 [10] C M Gordon, Y Q Wu, Toroidal and annular Dehn fillings, Proc. London Math. Soc. $$(3)$$ 78 (1999) 662 · Zbl 1024.57020 [11] C Hayashi, K Motegi, Only single twists on unknots can produce composite knots, Trans. Amer. Math. Soc. 349 (1997) 4465 · Zbl 0883.57005 [12] S Lee, S Oh, M Teragaito, Reducing Dehn fillings and small surfaces, Proc. London Math. Soc. $$(3)$$ 92 (2006) 203 · Zbl 1087.57013 [13] S Oh, Reducible and toroidal 3-manifolds obtained by Dehn fillings, Topology Appl. 75 (1997) 93 · Zbl 0870.57008 [14] S Oh, Reducing spheres and Klein bottles after Dehn fillings, Canad. Math. Bull. 46 (2003) 265 · Zbl 1028.57013 [15] T M Price, Homeomorphisms of quaternion space and projective planes in four space, J. Austral. Math. Soc. Ser. A 23 (1977) 112 · Zbl 0423.57006 [16] M Teragaito, Creating Klein bottles by surgery on knots, J. Knot Theory Ramifications 10 (2001) 781 · Zbl 1001.57039 [17] M Teragaito, Distance between toroidal surgeries on hyperbolic knots in the 3-sphere, Trans. Amer. Math. Soc. 358 (2006) 1051 · Zbl 1089.57007 [18] Y Q Wu, Dehn fillings producing reducible manifolds and toroidal manifolds, Topology 37 (1998) 95 · Zbl 0886.57012 [19] Y Q Wu, Sutured manifold hierarchies, essential laminations, and Dehn surgery, J. Differential Geom. 48 (1998) 407 · Zbl 0917.57015
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.