結び目の数学, 2008/12/24, 14:15–14:45 Seifert fibered surgeries on Montesinos knots ちょん い ん で In Dae Jong (鄭 仁大) Osaka City University (大阪市立大学) joint work with Kazuhiro Ichihara (Nara University of Education) and Shigeru Mizushima (Tokyo Institute of Technology) x1. Introduction ¶ Dehn surgery on a knot K in S3 ³ 1) Remove a neighborhood of K from S3. 2) Gluing a solid torus back (along slope γ). ´ γ = [f(m)] -15 x1. Introduction ¶ Dehn surgery on a knot K in S3 ³ 1) Remove a neighborhood of K from S3. 2) Gluing a solid torus back (along slope γ). ´ f γ m γ = [f(m)] -15-a By using the preferred meridian-longitude system, one can parameterize slopes by irr. fractions. i.e., fslopesg Ã!1:1 Q [ f1=0g For r 2 Q, r-surgery: surgery along a slope parameterized by r, K(r): the manifold obtained by r-surgery along a knot K. Today, we assume that all surgeries are non-trivial. Namely, we set aside the trivial(1=0-) surgery. -14 By using the preferred meridian-longitude system, one can parameterize slopes by irr. fractions. i.e., fslopesg Ã!1:1 Q [ f1=0g For r 2 Q, r-surgery: surgery along a slope parameterized by r. K(r): the manifold obtained by r-surgery along a knot K. Today, we assume that all surgeries are non-trivial. Namely, we set aside the trivial(1=0-) surgery. -14-a ¶ Problem ³ On hyperbolic knots in S3, determine all non- trivial Dehn surgeries producing non-hyperbolic 3- mfds. (Determine exceptional surgeries.) ´ Such surgeries are only finitely many. [Thurston] ¶ Types of exceptional surgeries ³ ² Reducible surgery ² Toroidal surgery ² Seifert fibered (SF) surgery ´ (as a consequence of Geometrization conjecture) -13 ¶ Today’s problem ³ Determine all exceptional surgeries on hyperbolic Montesinos knots. ´ ¶ Montesinos link M(R1;:::;Rl) ³ R1 R2 Rl l: length (Ri 2 Q $ a rational tangle). If Ri = 1=ai for all i (ai 2 Znf0g), then we denote it by P (a1; : : : ; al) a pretzel knot of type (a1; : : : ; al). ´ -12 ¶ Today’s problem ³ Determine all exceptional surgeries on hyperbolic Montesinos knots. ´ ¶ Montesinos link M(R1;:::;Rl) ³ R1 R2 Rl l: length (Ri 2 Q $ a rational tangle). If Ri = 1=ai for all i (ai 2 Znf0g), then we denote it by P (a1; : : : ; al) a pretzel knot of type (a1; : : : ; al). ´ -12-a ¶ Review ³ For a hyperbolic Montesinos knot with length l, ² l · 2 ) K is 2-bridge knot. All exceptional surgeries are classified [Brittenham-Wu ’95]. ² l ¸ 4 ) 6 9 exceptional surgery [Wu ’96]. ² 6 9 reducible surgery [Wu ’96]. ² All toroidal surgeries are classified [Wu ’06]. ´ Remains are SF surgeries with l = 3. Target: K = M(R1;R2;R3), K(r) : SF. -11 ¶ Review ³ For a hyperbolic Montesinos knot with length l, ² l · 2 ) K is 2-bridge knot. All exceptional surgeries are classified [Brittenham-Wu ’95]. ² l ¸ 4 ) 6 9 exceptional surgery [Wu ’96]. ² 6 9 reducible surgery [Wu ’96]. ² All toroidal surgeries are classified [Wu ’06]. ´ Remains are SF surgeries with l = 3. Target: K = M(R1;R2;R3), K(r) : SF. -11-a ¶ Review ³ For a hyperbolic Montesinos knot with length l, ² l · 2 ) K is 2-bridge knot. All exceptional surgeries are classified [Brittenham-Wu ’95]. ² l ¸ 4 ) 6 9 exceptional surgery [Wu ’96]. ² 6 9 reducible surgery [Wu ’96]. ² All toroidal surgeries are classified [Wu ’06]. ´ Remains are SF surgeries with l = 3. Target: K = M(R1;R2;R3), K(r) : SF. -11-b ¶ Review ³ For a hyperbolic Montesinos knot with length l, ² l · 2 ) K is 2-bridge knot. All exceptional surgeries are classified [Brittenham-Wu ’95]. ² l ¸ 4 ) 6 9 exceptional surgery [Wu ’96]. ² 6 9 reducible surgery [Wu ’96]. ² All toroidal surgeries are classified [Wu ’06]. ´ Remains are SF surgeries with l = 3. Target: K = M(R1;R2;R3), K(r) : SF. -11-c ¶ Review ³ For a hyperbolic Montesinos knot with length l, ² l · 2 ) K is 2-bridge knot. All exceptional surgeries are classified [Brittenham-Wu ’95]. ² l ¸ 4 ) 6 9 exceptional surgery [Wu ’96]. ² 6 9 reducible surgery [Wu ’96]. ² All toroidal surgeries are classified [Wu ’06]. ´ Remains are SF surgeries with l = 3. Target: K = M(R1;R2;R3), K(r) : SF. -11-d r-surgery on K is cyclic (resp. finite) = ¼1(K(r)) is cyclic (resp. finite). Result in the case that j¼1(K(r))j < 1 Theorem 1 [Ichihara-J.] (arXiv:0807.0905) K : a hyperbolic Montesinos knot. (i) If r-surgery on K is cyclic, then K = P (¡2; 3; 7) and r = 18 or 19. (ii) If r-surgery on K is acyclic finite, then K = P (¡2; 3; 7) and r = 17, or K = P (¡2; 3; 9) and r = 22 or 23. -10 Together with the result by [Wu], we have; Corollary Among hyperbolic arborescent knots, only the knots in Thm.1 admit such surgeries. ¶ ³ For p 2 f5; 7; ¢ ¢ ¢ ; 25g, the (¡2; p; p)-pretzel knot admit no finite surgeries. (by using Khovanov homology) (arXiv:0807.1341) ´ ¶ ³ A complete classification of finite surgeries on (¡2; p; q)-pretzel knots with p; q: odd positive. (arXiv:0809.4278) ´ -9 Together with the result by [Wu], we have; Corollary Among hyperbolic arborescent knots, only the knots in Thm.1 admit such surgeries. ¶ [Watson] ³ For p 2 f5; 7; ¢ ¢ ¢ ; 25g, the (¡2; p; p)-pretzel knot admit no finite surgeries. (by using Khovanov homology) (arXiv:0807.1341) ´ ¶ ³ A complete classification of finite surgeries on (¡2; p; q)-pretzel knots with p; q: odd positive. (arXiv:0809.4278) ´ -9-a Together with the result by [Wu], we have; Corollary Among hyperbolic arborescent knots, only the knots in Thm.1 admit such surgeries. ¶ [Watson] ³ For p 2 f5; 7; ¢ ¢ ¢ ; 25g, the (¡2; p; p)-pretzel knot admit no finite surgeries. (by using Khovanov homology) (arXiv:0807.1341) ´ ¶ [Futer-Ishikawa-Kabaya-Mattman-Shimokawa] ³ A complete classification of finite surgeries on (¡2; p; q)-pretzel knots with p; q: odd positive. (arXiv:0809.4278) ´ -9-b Result in the case that j¼1(K(r))j = 1 Theorem 2 [Ichihara-J.-Mizushima]. K: alternating hyperbolic Montesinos knot. If r-surgery on K is SF, then K = P (a; b; c) with a = 3 or 5 and a < b < c : odds. Key ingredients ² Essential (genuine) lamination ² Symmetries of knots (strong involution & cyclic period) ² The alternation number (The Rasmussen invariant, the signature, #-move, sharper Bennequin inequality, etc.) ² Hyperbolic structure (6-theorem) -8 x2 Proof of Theorem 1 Theorem 1 [Ichihara-J.] (arXiv:0807.0905) K : a hyperbolic Montesinos knot. (i) If r-surgery on K is a non-trivial cyclic, then K = P (¡2; 3; 7) and r = 18 or 19. (ii) If r-surgery on K is a non-trivial acyclic finite, then K = P (¡2; 3; 7) and r = 17, or K = P (¡2; 3; 9) and r = 22 or 23. Key ingredients ² Essential lamination ² Heegaard Floer homology -7 [Outline of Proof of Thm 1.] K : a hyperbolic Montesinos knot. ¶ Fact 1 [Delman] ³ If K admits a cyclic / finite surgery, then K is equivalent to either (i) P (¡2l; p; q), (ii) P (¡1; ¡1; 2m; p; q), or (iii) P (¡1; 2n; p; q). Here l; m ¸ 2, n =6 0, and 3 · p · q: odd. ´ [Delman] (unpublished, 1995). “Constructing essential laminations and taut foliations which survive all Dehn surgeries” -6 [Delman] Every hyperbolic Montesinos knot except for the families (i)–(iii) admits an essential lamination in its exterior which survives after all non-trivial Dehn surgeries (called persistent lamination). Essential lamination (introduced by [Gabai-Oertel]) Actually they showed that if a 3-mfd. M contains an essential lamination, then its universal cover must be R3. In particular, ¼1(M) is never cyclic/finite. ) We check the families (i)–(iii). -5 Case 1: P (¡2`; p; q) ¶ Fact 2 [Mattman] ³ If K admits a cyclic / finite surgery, then K =6 P (¡2l; p; q) with l ¸ 2 & 3 · p · q: odd. ´ [Mattman], “Cyclic and finite surgeries on pretzel knots”, J. Knot Theory Ramifications, 11(6):891–902, 2002. -4 Case 2: P (¡1; ¡1; 2m; p; q) ¶ Fact 3 [Ni] ³ If a knot in S3 admits a cyclic / finite surgery, then it must be a fibered knot. ´ ¶ Claim 1 ³ P (¡1; ¡1; 2m; p; q) is not fibered with m ¸ 2 and 3 · p · q: odd. ´ Claim 1 is shown by using an algorithm due to [Gabai], which decides fiberedness of a pretzel knot. -3 Case 2: P (¡1; ¡1; 2m; p; q) ¶ Fact 3 [Ni] ³ If a knot in S3 admits a cyclic / finite surgery, then it must be a fibered knot. ´ ¶ Claim 1 ³ P (¡1; ¡1; 2m; p; q) is not fibered with m ¸ 2 and 3 · p · q: odd. ´ Claim 1 is shown by using an algorithm due to [Gabai], which decides fiberedness of a pretzel knot. -3-a Case 3: P (¡1; 2n; p; q) ¶ L-space ³ A rational homology sphere Y is an L-space d if the rank of HF (Y ) is equal to jH1(Y ; Z)j. ´ In fact, ¼1(M) : cyclic / finite ) M : L-space . ¶ Fact 4 [Ozsv´ath-Szab´o] ³ A knot K in S3 admits an integral surgery yielding an L-space ) 8non-zero coeff. of ∆K(t) is §1. ´ ¶ Fact 5 ³ For a knot K ½ S3, K(p=q) is an L-space ) K(p) is also an L-space. ´ -2 Case 3: P (¡1; 2n; p; q) ¶ L-space ³ A rational homology sphere Y is an L-space d if the rank of HF (Y ) is equal to jH1(Y ; Z)j.