ON EXCEPTIONAL SURGERIES ON MONTESINOS KNOTS （モンテシノス結び目にそった例外的デーン手術について） KAZUHIRO ICHIHARA（市原 一裕）, IN DAE JONG（鄭 仁大）, AND SHIGERU MIZUSHIMA（水嶋 滋） Dedicated to Professor Kunio Murasugi on the occasion of his 80th birthday Abstract. In this article, we will report recent progresses of the sturdy on exceptional surgeries on Montesinos knots. In particular, we will consider toroidal Seifert fibered surgeries on non-pretzel Montesinos knots.
1. Introduction A Dehn surgery on a knot (i.e., an embedded simple closed curve) in the 3-sphere S 3 is defined as an operation to create a new 3-manifold as follows: Take an open tubular neighborhood of the knot, remove it, and glue a solid torus back. The resultant manifold via Dehn surgery is determined by the surgery slope on the peripheral torus T ; that is, the isotopy class of the curve on T identified with the meridian of the glued solid torus. Usually the set of slopes on T are identified with the set of irreducible fractions with 1/0 by using the standard meridian-longitude system. See [32] for example. Thus, we denote by K(r) the manifold obtained by Dehn surgery on a knot K in S 3 along the slope corresponding to a rational number r. One of the motivations to study Dehn surgery comes from the following famous fact, now called the Hyperbolic Dehn Surgery Theorem, due to W.P. Thurston [34, Theorem 5.8.2]: On a hyperbolic knot (i.e., a knot with hyperbolic complement), all but finitely many Dehn surgeries yield hyperbolic 3-manifolds. In view of this, such finitely many exceptions are called exceptional surgeries. Recently we are proceeding a study on exceptional surgeries on Montesinos knots. A Montesinos knot of type (R1 , R2 , . . . , Rl ), denoted by M (R1 , R2 , . . . , Rl ), is defined as a knot admitting a diagram obtained by putting rational tangles R1 , R2 , . . . , Rl together in a circle (see Figure 1). The minimal number of such rational tangles are called the length of the Montesinos knot. In particular, a Montesinos knot is called a (a1 , · · · , an )pretzel knot, denoted by P (a1 , · · · , an ), if the rational tangles contained in the knot are 1/a1 , · · · , 1/an . 2000 Mathematics Subject Classification. Primary 57M50; Secondary 57M25. Key words and phrases. exceptional surgery, Seifert fibered surgery, toroidal Seifert fibered surgery, Montesinos knot, alternating knot. Report manuscript for the Proceeding of “Intelligence of Low Dimensional Topology in honor of Professor Kunio Murasugi ’s 80th birthday ” (2009.11.12–14, Osaka City University).
Figure 1. Montesinos knot M (1/2, 1/3, −2/3) In this article, we report some recent results obtained by the authors or a part of authors. 1.1. Known facts. Before stating our results, in this subsection, we collect known facts about exceptional surgeries on Montesinos knots. First of all, it is already known which Montesinos knots are non-hyperbolic. A Montesinos knot of length at most two is in fact a two-bridge knot, and among two-bridge knots, non-hyperbolic ones are known to be the (2, n)-torus knots, shown by Menasco [22]. If a Montesinos knot K of length at least three is non-hyperbolic, then K is either P (−2, 3, 3) and P (−2, 3, 5), which are actually the torus knots of type (3, 4) or of type (3, 5). This was originally shown by Oertel [27, Corollary 5] together with the result in the unpublished monograph [2] by Bonahon and Siebenmann. Note that Dehn surgeries on torus knots are completely classified [23]. Thus, to study exceptional surgeries on Montesinos knots, we may exclude these knots above to consider. About exceptional surgeries on Montesinos knots of length two, which are in facts two-bridge knots, Brittenham and Wu studied extensively in [4], and gave a complete list of such surgeries. On the other hand, it was shown by Wu [37] that a Montesinos knot of length at least four admits no exceptional surgery. See also Wu [38, Section 4] for a survey. Therefore the remaining is the case where the length is just three. Furthermore, by virtue of the affirmative solution to the Geometrization Conjecture raised by W.P. Thurston in [35], established by Perelman [29, 30, 31], exceptional surgeries are classified into the following three types; • reducible surgery (yielding a reducible manifold) • toroidal surgery (yielding a toroidal manifold) • Seifert fibered surgery (yielding a Seifert fibered manifold) Here a closed orientable 3-manifold is called reducible / toroidal / Seifert fibered if it contains essential 2-spheres / it contains essential tori it admits a foliation by circles. On this classification, it was shown by Wu in [37, Corollary 2.6] that there are no reducible surgeries on Montesinos knots, and also given in [39] that a complete classification of toroidal surgeries on Montesinos knots.
Consequently, in the following, we will consider Seifert fibered surgeries on hyperbolic Montesinos knots of length three. 2. Results 2.1. Cyclic / Finite surgery. We first consider non-trivial Dehn surgery on a knot in S 3 which produce 3-manifolds with cyclic or finite fundamental groups, which we call cyclic surgeries / finite surgeries respectively. Now, by virtue of the affirmative solution to the Geometrization Conjecture, such surgeries are shown to be Seifert fibered surgeries, and expected to be very special. In fact, Culler, Gordon, Luecke and Shalen [6] (respectively, Boyer and Zhang [3]) proved there are at most three cyclic surgeries (resp., at most five finite surgeries). Furthermore, it is conjectured that knots admitting cyclic (resp., finite) surgeries are so-called “doubly primitive” (resp., “primitive/Seifert fibered”) knots, which were introduced by Berge [1] (resp., Dean [7]). See [20, Problem 1.77] for more information. Also it was shown by Delman and Roberts in [8] that no hyperbolic alternating knot admits a cyclic or finite surgery. One of the other well-known classes of knots, containing non-alternating ones, is given by the Montesinos knots, and, in fact, in [14], we show the following: Theorem 1. [14] Let K be a hyperbolic Montesinos knot. If K admits a non-trivial cyclic surgery, then K must be equivalent to the (−2, 3, 7)–pretzel knot and the surgery slope is 18 or 19. If K admits a non-trivial acyclic finite surgery, then K must be equivalent to either the (−2, 3, 7)–pretzel knot and the surgery slope is 17, or the (−2, 3, 9)–pretzel knot and the surgery slope is 22 or 23. We omit the details of the proof here. Please see [14] for details. Remark 1. Recently, using Khovanov homology, it was shown by Watson in [36, Theorem 7.5] that the (−2, p, p)–pretzel knot does not admit finite surgeries for p ∈ {5, 7, · · · , 25}. Also, very recently, Futer, Ishikawa, Kabaya, Mattman, and Shimokawa [10] obtained, independently, a complete classification of finite surgeries on (−2, p, q)–pretzel knots with odd positive integers p and q. 2.2. On alternating knots. Another class of knots, for which exceptional surgeries are also extensively studied, is given by that of alternating knots. It also includes that of two-bridge knots. For alternating Montesinos knots, we could prove the following: Theorem 2. [16] No alternating Montesinos knots admit Seifert fibered surgeries except for (2, n)-torus knots and twist knots. This follows from the following recent result obtained by Wu in [40]:
Theorem 3. [40] If M (p1 /q1 , p2 /q2 , p3 /q3 ) with q1 ≤ q2 ≤ q3 admits an atoroidal Seifert fibered surgery, then q1 = 2, (q1 , q2 ) = (3, 3), or (q1 , q2 , q3 ) = (3, 4, 5). On the other hand, for alternating case, we have already obtained; q1 , q2 , q3 are all odd and mutually distinct as follows: Theorem 4. Let K be an alternating hyperbolic Montesinos knot of length three. If K(r) is Seifert fibered, then K is equivalent to a pretzel knot P (p, q, r) for positive odd integers p, q, r with p < q < r. For this result, our paper is now in preparation. See [15] for an outline of the proof. Remark 2. It is known that a Montesinos knot is alternating if and only if its reduced Montesinos diagram (introduced in [21, Section 4]) is alternating. See [33, Section 4] and [21] for detail. 2.3. Toroidal Seifert surgery. As we noted before, each exceptional surgery is reducible, toroidal, or Seifert fibered. Here is a remark: They are NOT exclusive. i.e., there are non-empty intersections. However, F. Gonz`alez-Acu˜ na and H. Short conjectured in [11] that the only way to get a reducible 3-manifold by surgery on a knot in S 3 is to surger on a cable knot along the slope determined by the cabling annulus. This is now called the Cabling Conjecture, and still remaining open. See [20, Problem 1.79]. Thus the case which should be considered is Dehn surgery both toroidal and Seifert fibered, which we call toroidal Seifert fibered surgery. On toroidal Seifert fibered surgeries, there are some known results: There actually exists infinitely many hyperbolic knots in S 3 each of which admits a toroidal Seifert fibered surgery. They were found by Eudave-Mu˜ noz [9, Proposition 4.5(1) and (3)], and also by Gordon and Luecke [12] independently. On the other hand, Motegi studied in [24] toroidal Seifert fibered surgeries on periodic knots, and gave several restrictions on the existence of such surgeries. In particular, it was shown that only the trefoil admits toroidal Seifert fibered surgeries among two-bridge knots [24, Corollary 1.6]. Related results were also obtained by Patton in [28, Theorem 11]. Now, on alternating knots / Montesinos knots, we could have the following results: Theorem 5. [16] Other than trefoil and connected sums of torus knots, alternating knots admit no toroidal Seifert fibered surgeries. Theorem 6. [17] Non-pretzel Montesinos knots admit no toroidal Seifert fibered surgeries. In Theorem 6, the condition that “non-pretzel” would be removed. Please see our forthcoming paper (in preparation).
3. Outline of the Proof of Theorem 6 In this section, we give a sketch of proof of Theorem 6. Let K be a non-pretzel Montesinos knot. As we claimed, we may assume that K is hyperbolic and length is equal to three. Now suppose for contradiction that K(r) is toroidal Seifert fibered for some rational number r. Then we first have the following: Claim 1. The knot K must be a fibered knot and r = 0. This directly follows from [18, Proposition 1], and fact that K is small, i.e., the exterior of K contains no closed essential surface, shown by Oertel [27, Corollary 4(a)]. Together with this Claim 1, the classification of toroidal surgery on Montesinos knots given by [39] implies the following: Claim 2. The knot K must be M (−1/2, 1/3, 1/(6 ± 1/2)). It thus suffices to show the next two claims: Claim 3. K 6= M (−1/2, 1/3, 1/(6 + 1/2)) Proof. Suppose that K = M (−1/2, 1/3, 1/(6 + 1/2)) and K(0) is Seifert fibered. Since K is fibered, K(0) fibers over the circle with genus two fiber F . Then, by [19, VI. 31. Lemma], the monodromy map ϕ for K(0) must be periodic. Together with [5, Lemma 5.1], this implies that the Alexander polynomial ∆K (t) for K must have a root of unity as zeros. (Note that the characteristic polynomial for the isomorphism on H1 (F ) induced by ϕ is equal to ∆K (t).) However, by direct calculations for K = M (−1/2, 1/3, 1/(6+1/2)), ∆K (t) = −(t2 +t−2 )+3, which has no roots of unity as zeros. A contradiction occurs. ¤ Claim 4. K 6= M (−1/2, 1/3, 1/(6 − 1/2)) Proof. Suppose that K = M (−1/2, 1/3, 1/(6 − 1/2)) and K(0) is Seifert fibered. Then, by Kirby moves (c.f. [32] for example), we obtain a framed link presentation of the surface bundle K(0) illustrated as in Figure 2. From the figure, we can identify the monodromy map of K(0) is described as τ53 ◦ τ3 ◦ τ1−1 ◦ τ4 ◦ τ2−1 , where τi denotes the Dehn twist along the i-th Lickorish generator on the genus two surface F2 . See [13, Section 3] for example. Then we see that K(0) is the double-branched covering space of S 2 × S 1 with the closure of the braid β = σ53 σ3 σ1−1 σ4 σ2−1 as the branch set. Here σi denotes the standard i-th generator of the braid group B6 (S2 )). See [26]. Now it suffice to show that β is not finite order in B6 (S2 )). To see this, by [26, Proposition 2.3, Chap.11], we can use the exponent sum for braids. Actually exp(β) = 3 6≡ 0, 5 mod 10, which shows that β is not finite order in B6 (S2 )) by [25, Theorem 4.2]. ¤
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Figure 2. Kirby moves Acknowledgments The author is partially supported by Grant-in-Aid for Young Scientists (B), No. 18740038, Ministry of Education, Culture, Sports, Science and Technology, Japan. The authors would like to thank Makoto Sakuma for letting their know the result obtained by Professor Kunio Murasugi in [25]. References 1. J. Berge, Some knots with surgeries yielding lens spaces, unpublished manuscript. 2. F. Bonahon and L. Siebenmann, Geometric splittings of knots, and conway’s algebraic knots, Draft of a monograph (1979-85). 3. S. Boyer and X. Zhang, A proof of the finite filling conjecture, J. Differential Geom. 59 (2001), 87–176.
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[email protected] Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Ohokayama, Meguro, Tokyo 152-8552, Japan. （152-8552 東京都目黒区大岡山 2-12-1 東京工業大学大学院情報理工学研究科数理・計算科学専攻） E-mail address:
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