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172 W. MENASCO AND X. ZHANG by T1, T2, T3. In Figure 8 (a), we have chosen framings for the two inside 2-spheres S1 (the one on the left) and S2 (the one on the right). We may assume that T1 covers S1 and T2 covers S2. Framings on T1 and T2 are lifts of those on S1 and S2 respectively. Now we let Nn be the manifold obtained by Dehn filling M3 along T2 with slope 1/n. Then it follows from Thurston’s hyperbolic surgery theorem that the manifolds Nn’s are hyperbolic and are mutually different for infinitely many choices of n (because that in the core of the filling solid torus is a geodesic whose length becomes arbitrarily small as n becomes large). Each Nn has its boundary consisting of two tori and is the double branched cover of a once-punctured 3-ball Yn obtained by filling X along S2 with a 1/n-tangle, as shown in Figure 8 (b). Now Dehn filling Nn along T2 with the 0-slope will produces a Seifert fibered space (which is homeomorphic to the trefoil knot exterior in S 3 ) since the resulting manifold is the double branched cover of the tangle shown in Figure 8 (c), obtained by filling S1 with a 0-tangle. On the other hand, Dehn filling Nn along T2 with the −3/2-slope will produces a manifold which contains an essential torus and an essential annulus since the resulting manifold is the double branched cover of the tangle shown in Figure 8 (d), obtained by filling S1 with the −3/2-tangle. There remains unsettled the case concerning the optimal bound on the distance between two Seifert Dehn filling slopes on a torus boundary component T of a compact orientable simple manifold M with at least two boundary components. Namely what is the minimal upper bound on ∆(α, β) if both M(T, β) and M(T, α) are Seifert fibered manifolds? The best known bound is 3 obtained recently in [GW3]. Conjecture 18. The optimal bound is 2. The distance two can be realized on the Whitehead link exterior M; both the 1-slope and the 3-slope on any component of ∂M (with respect to the standard framings) produce Seifert fibered spaces. References [A] C. Adams, Toroidally alternating knots and links, Topology, 33 (1994), 353-369. [B] J. Berge, The knots in D 2 × S 1 with nontrivial Dehn surgeries yielding D 2 × S 1 , Topology Appl., 38 (1991), 1-19. [BZ] S. Boyer and X. Zhang, Reducing Dehn filling and toroidal Dehn filling, Topology Appl., 68 (1996), 285-303. [BZ2] , On Culler-Shalen seminorms and Dehn filling, Ann. Math., 148 (1998), 737-801. [CGLS] M. Culler, C.M. Gordon, J. Luecke and P. Shalen, Dehn surgery on knots, Ann. of Math., 125 (1987), 237-300.
TANGLES, 2-HANDLE ADDITIONS AND DEHN FILLINGS 173 [D] J. Dean, Hyperbolic knots with small Seifert-fibered Dehn surgeries, PhD Thesis, University of Texas at Austin (1996); (available at http://www.math.lsa. umich.edu/dean/research). [E] M. Eudave-Munoz, Band sum of links which yield composite links. The cabling conjecture for strongly invertible knots, Trans. Amer. Math. Soc., 330 (1992), 463-501. [E2] , On nonsimple 3-manifolds and 2-handle addition, Topology Appl., 55 (1994), 131-152. [EW] M. Eudave-Munoz and Y. Wu, Nonhyperbolic Dehn fillings on hyperbolic 3manifolds, Pacific J. Math., 190 (1999), 261-275. [Ga] D. Gabai, Foliations and the topology of 3-manifolds III, J. Diff. Geom., 26 (1987), 479-536. [Ga2] , 1-bridge braids in solid tori, Topology Appl., 37 (1990), 221-235. [GO] D. Gabai and U. Oertel, Essential laminations in 3-manifolds, Ann. Math., 130 (1989), 41-73. [GS] F. Gonzalez-Acuna and H. Short, Knot surgery and primeness, Math. Proc. Camb. Phil. Soc., 99 (1986), 89-102. [G] C. Gordon, Dehn surgery and satellite knots, Trans. Amer. Math. Soc., 275 (1983), 687-708. [GL] C. Gordon and J. Luecke, Reducible manifolds and Dehn surgery, Topology, 35 (1996), 385-409. [GL2] C. Gordon and J. Luecke, Only integral surgery can yield reducible manifolds, Math. Camb. Phil. Soc., 102 (1987), 94-101. [GL3] , Knots are determined by their complements, J. Amer. Math. Soc., 2 (1989), 371-415. [GL4] , unpublished. [GL5] , Toroidal and boundary-reducing Dehn fillings, Topology Appl., 93 (1999), 77-90. [GW] C. Gordon and Y. Wu, Annular and boundary reducing Dehn fillings, Topology, to appear; (available at http://www.math.uiowa.edu/wu/papers/papers.html). [GW2] , Toroidal and annular Dehn fillings, Proc. London Math. Soc., 78 (1999), 662-700. [GW3] , Annular Dehn fillings, preprint; (available at http://www.math. uiowa.edu/wu/papers/papers.html). [Ha] A. Hatcher, On the boundary curve of incompressible surfaces, Pacific J. Math., 99 (1982), 373-377. [HS] C. Hayashi and K. Shimokawa, Symmetric knots satisfy the cabling conjecture, Math. Proc. Camb. Phil. Soc., 123 (1998), 501-529. [H] J. Hempel, 3-Manifolds, Ann. Math. Studies, 86 (1976). [Ho] J. Hoffman, There are no strict great x-cycles after a reducing or P 2 surgery on a knot, J. of Knots and Its Ramification, 7 (1998), 549-569. [J] W. Jaco, Lectures on three-manifolds topology, CBMS Regional Conf. Ser. Math., 43 (1980).
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Pacific Journal of Mathematics Volu
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PACIFIC JOURNAL OF MATHEMATICS Vol.
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EIGENVALUES ASYMPTOTICS 3 (i) If pm
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EIGENVALUES ASYMPTOTICS 5 Corollary
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Thus, by the Itô formula, we have
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Therefore, K1(t) ≡ t 2−d/2 ≤
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EIGENVALUES ASYMPTOTICS 11 The chan
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EIGENVALUES ASYMPTOTICS 13 Here Res
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IMAGINARY QUADRATIC FIELDS 17 Table
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IMAGINARY QUADRATIC FIELDS 19 and t
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IMAGINARY QUADRATIC FIELDS 21 Propo
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〈σ 〈σ〉 〈σ, τ〉 2 〈1
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IMAGINARY QUADRATIC FIELDS 25 Then
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IMAGINARY QUADRATIC FIELDS 27 were
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IMAGINARY QUADRATIC FIELDS 29 Thus
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IMAGINARY QUADRATIC FIELDS 31 [3] ,
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34 CARINA BOYALLIAN spherical serie
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36 CARINA BOYALLIAN Here I G MAN (
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38 CARINA BOYALLIAN where k ≤ [ n
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40 CARINA BOYALLIAN The case G = Sp
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42 CARINA BOYALLIAN we can, since i
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44 CARINA BOYALLIAN and this formul
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46 CARINA BOYALLIAN Here, J(ν) int
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48 CARINA BOYALLIAN [V] D. Vogan, R
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50 ROBERT F. BROWN AND HELGA SCHIRM
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82 GILLES CARRON à l’infini s’
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84 GILLES CARRON isométriques au d
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86 GILLES CARRON Comme D est ellipt
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88 GILLES CARRON Réciproquement si
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90 GILLES CARRON où H est la proje
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92 GILLES CARRON 2.a. Exemples repo
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94 GILLES CARRON alors si σ ∈ C
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96 GILLES CARRON Ainsi s’il exist
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98 GILLES CARRON Proposition 2.7. N
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102 GILLES CARRON compact. De même
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104 GILLES CARRON 4. Application du
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106 GILLES CARRON Dans le cas où l
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BRAIDED OPERATOR COMMUTATORS 111 Re
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BRAIDED OPERATOR COMMUTATORS 115 3.
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Then, for 1 < k ≤ m + 1, we have
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