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Essential tangle decompositions of knots with tunnel number one tangles
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Essential tangle decompositions of knots with tunnel number one tangles Toshio Saito Department of Mathematics, Joetsu University of Education, 1 Yamayashiki, Joetsu 943-8512, Japan
a r t i c l e
i n f o
Article history: Received 30 December 2013 Received in revised form 6 February 2014 Accepted 6 February 2014 Available online xxxx
a b s t r a c t It is shown by Ozawa that a knot in the 3-sphere has a unique essential tangle decomposition if it admits an essential free 2-tangle decomposition. We show that Ozawa’s result cannot be generalized even if a knot admits an essential 2-tangle decomposition such that one of the decomposed tangles is of tunnel number one. © 2015 Elsevier B.V. All rights reserved.
MSC: 57M25 57N10 Keywords: Tangle Tunnel number Weak reduction
1. Introduction For a positive integer m, an m-tangle (B, T ) is defined to be a pair of a 3-ball B and mutually disjoint m arcs T properly embedded in B. Let K be a knot in the 3-sphere S 3 and P ⊂ S 3 a 2-sphere intersecting K in 2m points. Then P cuts S 3 into two 3-balls, say B1 and B2 . Since P intersects K in 2m points, we see that each Bi (i = 1, 2) intersects K in a collection of mutually disjoint m arcs, say Ti . Hence each (Bi , Ti ) is an m-tangle. The decomposition (B1 , T1 ) ∪P (B2 , T2 ) is called an m-tangle decomposition of (S 3 , K), where P = (P, P ∩K). We call P a tangle sphere or an m-tangle sphere. A tangle decomposition (B1 , T1 ) ∪P (B2 , T2 ) is said to be essential if P is incompressible in (S 3 , K). In previous work [3], the author defined tunnel number, denoted by tnl(·), of a tangle which is a natural generalization of tunnel number of a knot. See the next section for definitions. We here notice that a tangle (B, T ) is of tunnel number zero if and only if it is a free tangle, i.e., the exterior of T in B is homeomorphic to a handlebody. A tangle decomposition (B1 , T1 ) ∪P (B2 , T2 ) is said to be free if each (Bi , Ti ) is a free tangle. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2015.05.053 0166-8641/© 2015 Elsevier B.V. All rights reserved.
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Theorem 1.1. (Ozawa [2, Theorem 1.2]) Let K be a knot in S 3 . Suppose that P gives an essential free 2-tangle decomposition. Then any tangle sphere giving an essential tangle decomposition of (S 3 , K) is ambient isotopic to P. The following is merely a restatement of Theorem 1.1 by using the notation tnl(·). Theorem 1.2 (Restatement of Theorem 1.1). Let K be a knot in S 3 with an essential 2-tangle decomposition (B1 , T1 ) ∪P (B2 , T2 ) with tnl(Ti ) = 0 for each i = 1, 2. Then any tangle sphere giving an essential tangle decomposition of (S 3 , K) is ambient isotopic to P. In this paper, we show that Theorem 1.2 cannot be generalized even if a knot admits an essential 2-tangle decomposition such that one of the decomposed tangles is of tunnel number one. Theorem 1.3. For any non-negative integer n, there is a knot K in S 3 which satisfies the following: (1) (S 3 , K) admits an essential 2-tangle decomposition (B1 , T1 ) ∪P (B2 , T2 ) with tnl(Ti ) = 1 and tnl(Tj ) = n for (i, j) = (1, 2) or (2, 1). (2) (S 3 , K) admits another essential 2-tangle decomposition different from the above. 2. Definitions Throughout this paper, we work in the piecewise linear category. Let B be a sub-manifold of a manifold A. The notation Nbd(B; A) denotes a (closed) regular neighborhood of B in A. By Ext(B; A), we mean the exterior of B in A, i.e., Ext(B; A) = cl(A \ Nbd(B; A)), where cl(·) means the closure. The notation | · | indicates the number of connected components. Let M be a compact connected orientable 3-manifold with non-empty boundary. Let J be a 1-manifold properly embedded in M and F a surface properly embedded in M . Here, a surface means a connected compact 2-manifold. We always assume that a surface intersects J transversely. Set M = (M, J) and F = (F, F ∩ J). For convenience, we also call F a surface. A simple closed curve properly embedded in F \ J is said to be inessential in F if it bounds a disk in F intersecting J in at most one point. A simple closed curve properly embedded in F \ J is said to be essential in F if it is not inessential in F. A surface F is compressible in M if there is a disk D ⊂ M \ J such that D ∩ F = ∂D and ∂D is essential in F. Such a disk D is called a compressing disk of F. We say that F is incompressible in M if F is not compressible in M. A 3-manifold C is called a (genus g) compression body if there exists a closed surface F of genus g such that C is obtained from F × [0, 1] by attaching 2-handles along mutually disjoint loops in F × {0} and filling in some resulting 2-sphere boundary components with 3-handles. We denote F × {1} by ∂+ C and ∂C \ ∂+ C by ∂− C. A compression body C is called a handlebody if ∂− C = ∅. The triplet (C1 , C2 ; S) is called a (genus g) Heegaard splitting of M if C1 and C2 are (genus g) compression bodies with C1 ∪ C2 = M and C1 ∩ C2 = ∂+ C1 = ∂+ C2 = S. The Heegaard genus hg(M ) of M is the minimal integer g for which M admits a genus g Heegaard splitting. A simple arc γ properly embedded in a compression body C is said to be vertical if γ is isotopic to an arc with {a point} × [0, 1] ⊂ ∂− C × [0, 1] relative to boundary. A simple arc γ properly embedded in C is said to be trivial if there is a disk δ in C with γ ⊂ ∂δ and ∂δ \ γ ⊂ ∂+ C. Such a disk δ is called a bridge disk of γ. A disjoint union of trivial arcs is said to be mutually trivial if they admit a disjoint union of bridge disks. We now recall definitions of a c-compression body and a c-Heegaard splitting given by Tomova [4]. Let J be a 1-manifold properly embedded in a compact connected orientable 3-manifold M with non-empty boundary. A surface F = (F, F ∩ J) is c-compressible in M = (M, J) if there is a disk D ⊂ M \ J such
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that D ∩ F = ∂D, ∂D is essential in F and D intersects J in at most one point. If |D ∩ J| = 1, then D is called a cut disk of F. We say that F is c-incompressible in M if F is not c-compressible in M. A c-disk is a compressing disk or a cut disk. Let C be a pair of a genus g compression body C and a 1-manifold J properly embedded in C. Then C = (C, J) is called a (genus g) c-compression body if there is a disjoint union D of c-disks and bridge disks which cuts C into some 3-balls and a 3-manifold homeomorphic to ∂− C × [0, 1] with vertical arcs. Then D is called a complete c-disk system of C. If D contains a compressing disk, then C is said to be compressible. We set ∂± C = (∂± C, ∂± C ∩ J). Definition 2.1. Let J be a 1-manifold properly embedded in a compact connected orientable 3-manifold M . The triplet (C1 , C2 ; S) is a (genus g) c-Heegaard splitting of M = (M, J) if C1 and C2 are (genus g) c-compression bodies with C1 ∪ C2 = M and C1 ∩ C2 = ∂+ C1 = ∂+ C2 = S. The surface S is called a c-Heegaard surface of M. ∼ S 2 and T a 1-manifold properly Let M be a compact connected orientable 3-manifold with ∂M = embedded in M . We say that (M, T ) is an m-tangle if T consists of m arcs. An m-tangle (M, T ) is said to be essential if the surface (∂M, ∂M ∩ T ) is incompressible in (M, T ). An n-tangle (M, T ) is said to be free if Ext(T ; M ) is a handlebody. Let (C1 , C2 ; S) be a c-Heegaard splitting of an m-tangle (M, T ) with ∂M = ∂− Ci for i = 1 or 2, say i = 2, where Ci = (Ci , Ci ∩ T ) and S = (S, S ∩ T ). Then we notice that C1 is a handlebody and C1 ∩ T consists of mutually trivial arcs. Such a c-Heegaard splitting (C1 , C2 ; S) is called a (g, b, c)-splitting of (M, T ), where g is the genus of the closed surface S, b is the number of trivial arcs C1 ∩ T and c is the number of the components of T each of which is contained in C2 . Definition 2.2. Let (M, T ) be an m-tangle. A disjoint union of simple arcs τ = τ1 ∪· · ·∪τm properly embedded in Ext(T ; M ) is called an unknotting tunnel system if cl(Ext(T ; M ) \ Nbd(τ ; M )) is a handlebody. The tunnel number tnl(T ) of (M, T ) is the minimal number of components of such unknotting tunnel systems. In particular, we define tnl(T ) = 0 if (M, T ) is a free tangle. Definition 2.3. Let K be a knot in a closed connected orientable 3-manifold M and P ⊂ M a separating 2-sphere which intersects K transversely in 2m(> 0) points. Then P cuts M into two 3-manifolds M1 and M2 so that (Mi , Ti ) (i = 1, 2) are m-tangles, where Ti = Mi ∩K. The decomposition (M1 , T1 )∪P (M2 , T2 ) is called an m-tangle decomposition, or a tangle decomposition for short. A tangle decomposition (M1 , T1 ) ∪P (M2 , T2 ) is said to be essential if each tangle (Mi , Ti ) is essential. 3. c-Weak reduction For an integer g > 2, let (C1 , C2 ; S) be a (g, 0, 2)-splitting of a 2-tangle (B, T ), where B is a 3-ball. By definition, we may assume that C1 = (C1 , ∅) where C1 is a handlebody of genus g and that C2 = (C2 , T ) where C2 is a compression body with ∂− C2 = ∂B. Let D1 and D2 be a collection of mutually disjoint disks in C1 and C2 respectively which satisfy the following three conditions: Assumption 3.1. (1) D1 consists of g − 2 compressing disks such that D1 cuts C1 into a genus two handlebody. (2) D2 consists of two cut disks such that D2 cuts C2 into a compression body of genus g − 2 with four vertical arcs. (3) ∂D1 ∩ ∂D2 = ∅.
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Fig. 1. c-Weak reduction of (C1 , C2 ; S).
Let (B, T ), (C1 , C2 ; S), D1 and D2 be above with Assumption 3.1. Then (B, T ) is decomposed into two c-Heegaard splittings by c-weak reduction treated in [4] as follows. Set Cii = cl(Ci \ Nbd(Di ; Ci )) for each i = 1, 2. Let Cij be a 3-manifold obtained by attaching Nbd(Dj ; Cj ) to ∂+ Ci × [0, 1] for each (i, j) = (1, 2), (2, 1). We notice that each Cij is a c-compression body. Hence we have (B, T ) = (C11 ∪ C12 ) ∪ (C21 ∪ C22 ), and Ci1 ∪ Ci2 is a c-Heegaard splitting for each i = 1, 2 (cf. Fig. 1). The inverse operation of c-weak reduction is called an amalgamation. Proposition 3.2. Let (B, T ), (C1 , C2 ; S), D1 and D2 be above with Assumption 3.1. Let P be a closed orientable surface obtained by c-weak reduction with respect to (D1, D2 ). Then P cuts a free tangle (B , T ) from (B, T ). Moreover, if (B , T ) is non-trivial, then P is incompressible in (B, T ). Proof. The surface P splits (B, T ) into two pieces (B , T ) and (B , T ) such that (B , T ) admits a (2, 0, 2)-splitting and that (B , T ) admits a c-Heegaard splitting illustrated in Fig. 1. Hence (B , T ) is a free 2-tangle. We now suppose that (B , T ) is non-trivial to have the second conclusion and suppose further that P is compressible in (B, T ) for a contradiction. If P is compressible in (B , T ), then (B , T ) is trivial since (B , T ) is a free tangle. Otherwise, P is compressible in Ext(B ; B). However, this is impossible since Ext(B ; B) is homeomorphic to S 2 × [0, 1] and Ext(B , B) ∩ T consists of four arcs each joins distinct boundary components. 2 Corollary 3.3. Let K be a knot in S 3 with an essential 2-tangle decomposition (B1 , T1 ) ∪P (B2 , T2 ). Suppose that (Bi , Ti ) (i = 1 or 2), say (B1 , T1 ), is of tunnel number greater than zero and that (C1 , C2 ; S) is a (g, 0, 2)-splitting of (B1 , T1 ) with a disk pair (D1 , D2 ) satisfying Assumption 3.1. Let P be a closed orientable surface obtained by c-weak reduction with respect to (D1 , D2 ). Then P cuts a free tangle (B1 , T1 ) from (B1 , T1 ). Moreover, if (B1 , T1 ) is non-trivial, then P gives an essential 2-tangle decomposition of (S 3 , K). Proof. Recall that P is a pair of a 2-sphere P and four points P ∩ K. Suppose that P does not give an essential 2-tangle decomposition of (S 3 , K). Then P is compressible in (S 3 , K). Let D be a compressing disk of P . If D ∩ P = ∅, then an innermost disk of D ∩ P in D must be removed by an ambient isotopy since P is incompressible in (S 3 , K). Hence we may assume that D is contained in (B1 , T1 ), i.e., P is compressible in (B1 , T1 ). Then we have the conclusion by the same argument in the proof of Proposition 3.2. 2 4. Construction of examples In this section, we will prove Theorem 1.3 by giving a concrete example. It will be enough to obtain a ˜ be a knot in S 3 such 2-tangle with a (3, 0, 2)-splitting satisfying Assumption 3.1. To this end, we first let K
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˜ Fig. 2. (B1 , T1 ) obtained from (S 3 , K).
˜ admits a (1, 2)-bridge splitting (V1 , V2 ; F). Namely, Vi is a pair of a solid torus Vi and two trivial that (S 3 , K) ˜ ˜ Let γi (i = 1, 2) be a “straight” arcs Vi ∩ K for each i = 1, 2 and F is a pair of a torus F and four points F ∩ K. ˜ arc joining two arc components in Vi which satisfies the following. Set Bi = (Nbd(γi ; Vi ), Nbd(γi ; Vi ) ∩ K) and Pi = ∂Bi for each i = 1, 2. We notice that each Bi is a trivial 2-tangle. We take each arc γi such that Vi ˜ removed the interior of Bi , say Vi , is a pair of a once punctured solid torus Vi and four vertical arcs Vi ∩ K (cf. Fig. 2). Let (B1 , T1 ) be a 2-tangle obtained from V1 ∪ V2 by capping off an essential free 2-tangle, say (B0 , T0 ), along P1 . Then we see that (B1 , T1 ) has the decomposition illustrated in Fig. 2. ˜ Pi , (B0 , T0 ) and (B1 , T1 ) above: Lemma 4.1. The following holds for K, (1) Each Pi (i = 1, 2) is incompressible in (B1 , T1 ). ˜ is not a 2-bridge knot, (2) If K (a) P1 is not parallel to P2 . (b) tnl(T1 ) = 1. Proof. Since (B1 , T1 ) has the decomposition illustrated in Fig. 2, amalgamating the decomposition shows that (B1 , T1 ) admits a (3, 0, 2)-splitting with Assumption 3.1. Hence P1 is incompressible in (B1 , T1 ) by Proposition 3.2. We also see that P2 is incompressible in (B1 , T1 ) by an argument similar to that in the ˜ is not a 2-bridge proof of Corollary 3.3 and hence we have the conclusion (1). We now assume that K knot. If P1 is parallel to P2 , then V1 ∪ V2 has a product structure, i.e., it is a pair of S 2 × [0, 1] and ˜ = B1 ∪ (V ∪ V ) ∪ B2 and each Bi is a trivial 2-tangle, we see that K ˜ four vertical arcs. Since (S 3 , K) 1 2 is a 2-bridge knot, a contradiction. Therefore P1 is not parallel to P2 . Finally we show tnl(T1 ) = 1. It is easy to see that tnl(T1 ) ≤ 1 since (B1 , T1 ) admits a (3, 0, 2)-splitting. Suppose for a contradiction that tnl(T1 ) = 0, i.e., (B1 , T1 ) is a free tangle. Let K be a knot in S 3 obtained by attaching two copies of (B1 , T1 ) along their boundaries. Then P2 gives an essential free 2-tangle decomposition of (S 3 , K ). It follows from Corollary 3.3 that P1 also gives an essential 2-tangle decomposition of (S 3 , K). The conclusion (2)(a) of this lemma indicates that (S 3 , K) admits distinct essential 2-tangle decompositions. This contradicts Ozawa’s uniqueness theorem of essential 2-tangle decompositions under the assumption that a knot has an essential free 2-tangle decomposition [2]. Hence tnl(T1 ) ≥ 1 and we have tnl(T1 ) = 1. 2 Remark 4.2. For example, Morimoto–Sakuma–Yokota knot in [1] admits a (1, 2)-splitting and is not a 2-bridge knot.
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Let (B2 , T2 ) be any essential 2-tangle and K a knot obtained by attaching (B1 , T1 ) with (B2 , T2 ) along P2 . Then (S 3 , K) admits an essential 2-tangle decomposition (B1 , T1 ) ∪P2 (B2 , T2 ) with tnl(T1 ) = 1. Let (B3 , T3 ) be the exterior of (B0 , T0 ) in (S 3 , K). Since (B3 , T3 ) is a 2-tangle, we have a tangle decomposition (B0 , T0 ) ∪P1 (B3 , T3 ). Hence we have: (B3 ,T3 )
(S 3 , K) = (B0 , T0 ) ∪ (V1 ∪ V2 ) ∪ (B2 , T2 ) . (B1 ,T1 )
It follows from Corollary 3.3 and Lemma 4.1 that P1 gives an essential 2-tangle decomposition of (S 3 , K) and is not parallel to P2 . Therefore we have the desired example if we take (B2 , T2 ) as an appropriate 2-tangle. Acknowledgements This work was supported by JSPS KAKENHI Grant Number 24740041. The author would like to thank the referee for useful comments. References [1] K. Morimoto, M. Sakuma, Y. Yokota, Examples of tunnel number one knots which have the property “1 + 1 = 3”, Math. Proc. Camb. Philos. Soc. 119 (1) (1996) 113–118. [2] M. Ozawa, On uniqueness of essential tangle decompositions of knots with free tangle decompositions, in: G.T. Jin, K.H. Ko (Eds.), Proc. Appl. Math. Workshop, vol. 8, KAIST, Taejon, 1998, pp. 227–232. [3] T. Saito, Tunnel number of tangles and knots, J. Math. Soc. Jpn. 66 (4) (2014) 1303–1313. [4] M. Tomova, Thin position for knots in a 3-manifold, J. Lond. Math. Soc. 80 (2009) 85–98.
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Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Essential tangle decompositions of knots with tunnel number one tangles Toshio Saito Department of Mathematics, Joetsu University of Education, 1 Yamayashiki, Joetsu 943-8512, Japan
a r t i c l e
i n f o
Article history: Received 30 December 2013 Received in revised form 6 February 2014 Accepted 6 February 2014 Available online xxxx
a b s t r a c t It is shown by Ozawa that a knot in the 3-sphere has a unique essential tangle decomposition if it admits an essential free 2-tangle decomposition. We show that Ozawa’s result cannot be generalized even if a knot admits an essential 2-tangle decomposition such that one of the decomposed tangles is of tunnel number one. © 2015 Elsevier B.V. All rights reserved.
MSC: 57M25 57N10 Keywords: Tangle Tunnel number Weak reduction
1. Introduction For a positive integer m, an m-tangle (B, T ) is defined to be a pair of a 3-ball B and mutually disjoint m arcs T properly embedded in B. Let K be a knot in the 3-sphere S 3 and P ⊂ S 3 a 2-sphere intersecting K in 2m points. Then P cuts S 3 into two 3-balls, say B1 and B2 . Since P intersects K in 2m points, we see that each Bi (i = 1, 2) intersects K in a collection of mutually disjoint m arcs, say Ti . Hence each (Bi , Ti ) is an m-tangle. The decomposition (B1 , T1 ) ∪P (B2 , T2 ) is called an m-tangle decomposition of (S 3 , K), where P = (P, P ∩K). We call P a tangle sphere or an m-tangle sphere. A tangle decomposition (B1 , T1 ) ∪P (B2 , T2 ) is said to be essential if P is incompressible in (S 3 , K). In previous work [3], the author defined tunnel number, denoted by tnl(·), of a tangle which is a natural generalization of tunnel number of a knot. See the next section for definitions. We here notice that a tangle (B, T ) is of tunnel number zero if and only if it is a free tangle, i.e., the exterior of T in B is homeomorphic to a handlebody. A tangle decomposition (B1 , T1 ) ∪P (B2 , T2 ) is said to be free if each (Bi , Ti ) is a free tangle. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2015.05.053 0166-8641/© 2015 Elsevier B.V. All rights reserved.
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Theorem 1.1. (Ozawa [2, Theorem 1.2]) Let K be a knot in S 3 . Suppose that P gives an essential free 2-tangle decomposition. Then any tangle sphere giving an essential tangle decomposition of (S 3 , K) is ambient isotopic to P. The following is merely a restatement of Theorem 1.1 by using the notation tnl(·). Theorem 1.2 (Restatement of Theorem 1.1). Let K be a knot in S 3 with an essential 2-tangle decomposition (B1 , T1 ) ∪P (B2 , T2 ) with tnl(Ti ) = 0 for each i = 1, 2. Then any tangle sphere giving an essential tangle decomposition of (S 3 , K) is ambient isotopic to P. In this paper, we show that Theorem 1.2 cannot be generalized even if a knot admits an essential 2-tangle decomposition such that one of the decomposed tangles is of tunnel number one. Theorem 1.3. For any non-negative integer n, there is a knot K in S 3 which satisfies the following: (1) (S 3 , K) admits an essential 2-tangle decomposition (B1 , T1 ) ∪P (B2 , T2 ) with tnl(Ti ) = 1 and tnl(Tj ) = n for (i, j) = (1, 2) or (2, 1). (2) (S 3 , K) admits another essential 2-tangle decomposition different from the above. 2. Definitions Throughout this paper, we work in the piecewise linear category. Let B be a sub-manifold of a manifold A. The notation Nbd(B; A) denotes a (closed) regular neighborhood of B in A. By Ext(B; A), we mean the exterior of B in A, i.e., Ext(B; A) = cl(A \ Nbd(B; A)), where cl(·) means the closure. The notation | · | indicates the number of connected components. Let M be a compact connected orientable 3-manifold with non-empty boundary. Let J be a 1-manifold properly embedded in M and F a surface properly embedded in M . Here, a surface means a connected compact 2-manifold. We always assume that a surface intersects J transversely. Set M = (M, J) and F = (F, F ∩ J). For convenience, we also call F a surface. A simple closed curve properly embedded in F \ J is said to be inessential in F if it bounds a disk in F intersecting J in at most one point. A simple closed curve properly embedded in F \ J is said to be essential in F if it is not inessential in F. A surface F is compressible in M if there is a disk D ⊂ M \ J such that D ∩ F = ∂D and ∂D is essential in F. Such a disk D is called a compressing disk of F. We say that F is incompressible in M if F is not compressible in M. A 3-manifold C is called a (genus g) compression body if there exists a closed surface F of genus g such that C is obtained from F × [0, 1] by attaching 2-handles along mutually disjoint loops in F × {0} and filling in some resulting 2-sphere boundary components with 3-handles. We denote F × {1} by ∂+ C and ∂C \ ∂+ C by ∂− C. A compression body C is called a handlebody if ∂− C = ∅. The triplet (C1 , C2 ; S) is called a (genus g) Heegaard splitting of M if C1 and C2 are (genus g) compression bodies with C1 ∪ C2 = M and C1 ∩ C2 = ∂+ C1 = ∂+ C2 = S. The Heegaard genus hg(M ) of M is the minimal integer g for which M admits a genus g Heegaard splitting. A simple arc γ properly embedded in a compression body C is said to be vertical if γ is isotopic to an arc with {a point} × [0, 1] ⊂ ∂− C × [0, 1] relative to boundary. A simple arc γ properly embedded in C is said to be trivial if there is a disk δ in C with γ ⊂ ∂δ and ∂δ \ γ ⊂ ∂+ C. Such a disk δ is called a bridge disk of γ. A disjoint union of trivial arcs is said to be mutually trivial if they admit a disjoint union of bridge disks. We now recall definitions of a c-compression body and a c-Heegaard splitting given by Tomova [4]. Let J be a 1-manifold properly embedded in a compact connected orientable 3-manifold M with non-empty boundary. A surface F = (F, F ∩ J) is c-compressible in M = (M, J) if there is a disk D ⊂ M \ J such
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that D ∩ F = ∂D, ∂D is essential in F and D intersects J in at most one point. If |D ∩ J| = 1, then D is called a cut disk of F. We say that F is c-incompressible in M if F is not c-compressible in M. A c-disk is a compressing disk or a cut disk. Let C be a pair of a genus g compression body C and a 1-manifold J properly embedded in C. Then C = (C, J) is called a (genus g) c-compression body if there is a disjoint union D of c-disks and bridge disks which cuts C into some 3-balls and a 3-manifold homeomorphic to ∂− C × [0, 1] with vertical arcs. Then D is called a complete c-disk system of C. If D contains a compressing disk, then C is said to be compressible. We set ∂± C = (∂± C, ∂± C ∩ J). Definition 2.1. Let J be a 1-manifold properly embedded in a compact connected orientable 3-manifold M . The triplet (C1 , C2 ; S) is a (genus g) c-Heegaard splitting of M = (M, J) if C1 and C2 are (genus g) c-compression bodies with C1 ∪ C2 = M and C1 ∩ C2 = ∂+ C1 = ∂+ C2 = S. The surface S is called a c-Heegaard surface of M. ∼ S 2 and T a 1-manifold properly Let M be a compact connected orientable 3-manifold with ∂M = embedded in M . We say that (M, T ) is an m-tangle if T consists of m arcs. An m-tangle (M, T ) is said to be essential if the surface (∂M, ∂M ∩ T ) is incompressible in (M, T ). An n-tangle (M, T ) is said to be free if Ext(T ; M ) is a handlebody. Let (C1 , C2 ; S) be a c-Heegaard splitting of an m-tangle (M, T ) with ∂M = ∂− Ci for i = 1 or 2, say i = 2, where Ci = (Ci , Ci ∩ T ) and S = (S, S ∩ T ). Then we notice that C1 is a handlebody and C1 ∩ T consists of mutually trivial arcs. Such a c-Heegaard splitting (C1 , C2 ; S) is called a (g, b, c)-splitting of (M, T ), where g is the genus of the closed surface S, b is the number of trivial arcs C1 ∩ T and c is the number of the components of T each of which is contained in C2 . Definition 2.2. Let (M, T ) be an m-tangle. A disjoint union of simple arcs τ = τ1 ∪· · ·∪τm properly embedded in Ext(T ; M ) is called an unknotting tunnel system if cl(Ext(T ; M ) \ Nbd(τ ; M )) is a handlebody. The tunnel number tnl(T ) of (M, T ) is the minimal number of components of such unknotting tunnel systems. In particular, we define tnl(T ) = 0 if (M, T ) is a free tangle. Definition 2.3. Let K be a knot in a closed connected orientable 3-manifold M and P ⊂ M a separating 2-sphere which intersects K transversely in 2m(> 0) points. Then P cuts M into two 3-manifolds M1 and M2 so that (Mi , Ti ) (i = 1, 2) are m-tangles, where Ti = Mi ∩K. The decomposition (M1 , T1 )∪P (M2 , T2 ) is called an m-tangle decomposition, or a tangle decomposition for short. A tangle decomposition (M1 , T1 ) ∪P (M2 , T2 ) is said to be essential if each tangle (Mi , Ti ) is essential. 3. c-Weak reduction For an integer g > 2, let (C1 , C2 ; S) be a (g, 0, 2)-splitting of a 2-tangle (B, T ), where B is a 3-ball. By definition, we may assume that C1 = (C1 , ∅) where C1 is a handlebody of genus g and that C2 = (C2 , T ) where C2 is a compression body with ∂− C2 = ∂B. Let D1 and D2 be a collection of mutually disjoint disks in C1 and C2 respectively which satisfy the following three conditions: Assumption 3.1. (1) D1 consists of g − 2 compressing disks such that D1 cuts C1 into a genus two handlebody. (2) D2 consists of two cut disks such that D2 cuts C2 into a compression body of genus g − 2 with four vertical arcs. (3) ∂D1 ∩ ∂D2 = ∅.
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Fig. 1. c-Weak reduction of (C1 , C2 ; S).
Let (B, T ), (C1 , C2 ; S), D1 and D2 be above with Assumption 3.1. Then (B, T ) is decomposed into two c-Heegaard splittings by c-weak reduction treated in [4] as follows. Set Cii = cl(Ci \ Nbd(Di ; Ci )) for each i = 1, 2. Let Cij be a 3-manifold obtained by attaching Nbd(Dj ; Cj ) to ∂+ Ci × [0, 1] for each (i, j) = (1, 2), (2, 1). We notice that each Cij is a c-compression body. Hence we have (B, T ) = (C11 ∪ C12 ) ∪ (C21 ∪ C22 ), and Ci1 ∪ Ci2 is a c-Heegaard splitting for each i = 1, 2 (cf. Fig. 1). The inverse operation of c-weak reduction is called an amalgamation. Proposition 3.2. Let (B, T ), (C1 , C2 ; S), D1 and D2 be above with Assumption 3.1. Let P be a closed orientable surface obtained by c-weak reduction with respect to (D1, D2 ). Then P cuts a free tangle (B , T ) from (B, T ). Moreover, if (B , T ) is non-trivial, then P is incompressible in (B, T ). Proof. The surface P splits (B, T ) into two pieces (B , T ) and (B , T ) such that (B , T ) admits a (2, 0, 2)-splitting and that (B , T ) admits a c-Heegaard splitting illustrated in Fig. 1. Hence (B , T ) is a free 2-tangle. We now suppose that (B , T ) is non-trivial to have the second conclusion and suppose further that P is compressible in (B, T ) for a contradiction. If P is compressible in (B , T ), then (B , T ) is trivial since (B , T ) is a free tangle. Otherwise, P is compressible in Ext(B ; B). However, this is impossible since Ext(B ; B) is homeomorphic to S 2 × [0, 1] and Ext(B , B) ∩ T consists of four arcs each joins distinct boundary components. 2 Corollary 3.3. Let K be a knot in S 3 with an essential 2-tangle decomposition (B1 , T1 ) ∪P (B2 , T2 ). Suppose that (Bi , Ti ) (i = 1 or 2), say (B1 , T1 ), is of tunnel number greater than zero and that (C1 , C2 ; S) is a (g, 0, 2)-splitting of (B1 , T1 ) with a disk pair (D1 , D2 ) satisfying Assumption 3.1. Let P be a closed orientable surface obtained by c-weak reduction with respect to (D1 , D2 ). Then P cuts a free tangle (B1 , T1 ) from (B1 , T1 ). Moreover, if (B1 , T1 ) is non-trivial, then P gives an essential 2-tangle decomposition of (S 3 , K). Proof. Recall that P is a pair of a 2-sphere P and four points P ∩ K. Suppose that P does not give an essential 2-tangle decomposition of (S 3 , K). Then P is compressible in (S 3 , K). Let D be a compressing disk of P . If D ∩ P = ∅, then an innermost disk of D ∩ P in D must be removed by an ambient isotopy since P is incompressible in (S 3 , K). Hence we may assume that D is contained in (B1 , T1 ), i.e., P is compressible in (B1 , T1 ). Then we have the conclusion by the same argument in the proof of Proposition 3.2. 2 4. Construction of examples In this section, we will prove Theorem 1.3 by giving a concrete example. It will be enough to obtain a ˜ be a knot in S 3 such 2-tangle with a (3, 0, 2)-splitting satisfying Assumption 3.1. To this end, we first let K
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˜ Fig. 2. (B1 , T1 ) obtained from (S 3 , K).
˜ admits a (1, 2)-bridge splitting (V1 , V2 ; F). Namely, Vi is a pair of a solid torus Vi and two trivial that (S 3 , K) ˜ ˜ Let γi (i = 1, 2) be a “straight” arcs Vi ∩ K for each i = 1, 2 and F is a pair of a torus F and four points F ∩ K. ˜ arc joining two arc components in Vi which satisfies the following. Set Bi = (Nbd(γi ; Vi ), Nbd(γi ; Vi ) ∩ K) and Pi = ∂Bi for each i = 1, 2. We notice that each Bi is a trivial 2-tangle. We take each arc γi such that Vi ˜ removed the interior of Bi , say Vi , is a pair of a once punctured solid torus Vi and four vertical arcs Vi ∩ K (cf. Fig. 2). Let (B1 , T1 ) be a 2-tangle obtained from V1 ∪ V2 by capping off an essential free 2-tangle, say (B0 , T0 ), along P1 . Then we see that (B1 , T1 ) has the decomposition illustrated in Fig. 2. ˜ Pi , (B0 , T0 ) and (B1 , T1 ) above: Lemma 4.1. The following holds for K, (1) Each Pi (i = 1, 2) is incompressible in (B1 , T1 ). ˜ is not a 2-bridge knot, (2) If K (a) P1 is not parallel to P2 . (b) tnl(T1 ) = 1. Proof. Since (B1 , T1 ) has the decomposition illustrated in Fig. 2, amalgamating the decomposition shows that (B1 , T1 ) admits a (3, 0, 2)-splitting with Assumption 3.1. Hence P1 is incompressible in (B1 , T1 ) by Proposition 3.2. We also see that P2 is incompressible in (B1 , T1 ) by an argument similar to that in the ˜ is not a 2-bridge proof of Corollary 3.3 and hence we have the conclusion (1). We now assume that K knot. If P1 is parallel to P2 , then V1 ∪ V2 has a product structure, i.e., it is a pair of S 2 × [0, 1] and ˜ = B1 ∪ (V ∪ V ) ∪ B2 and each Bi is a trivial 2-tangle, we see that K ˜ four vertical arcs. Since (S 3 , K) 1 2 is a 2-bridge knot, a contradiction. Therefore P1 is not parallel to P2 . Finally we show tnl(T1 ) = 1. It is easy to see that tnl(T1 ) ≤ 1 since (B1 , T1 ) admits a (3, 0, 2)-splitting. Suppose for a contradiction that tnl(T1 ) = 0, i.e., (B1 , T1 ) is a free tangle. Let K be a knot in S 3 obtained by attaching two copies of (B1 , T1 ) along their boundaries. Then P2 gives an essential free 2-tangle decomposition of (S 3 , K ). It follows from Corollary 3.3 that P1 also gives an essential 2-tangle decomposition of (S 3 , K). The conclusion (2)(a) of this lemma indicates that (S 3 , K) admits distinct essential 2-tangle decompositions. This contradicts Ozawa’s uniqueness theorem of essential 2-tangle decompositions under the assumption that a knot has an essential free 2-tangle decomposition [2]. Hence tnl(T1 ) ≥ 1 and we have tnl(T1 ) = 1. 2 Remark 4.2. For example, Morimoto–Sakuma–Yokota knot in [1] admits a (1, 2)-splitting and is not a 2-bridge knot.
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Let (B2 , T2 ) be any essential 2-tangle and K a knot obtained by attaching (B1 , T1 ) with (B2 , T2 ) along P2 . Then (S 3 , K) admits an essential 2-tangle decomposition (B1 , T1 ) ∪P2 (B2 , T2 ) with tnl(T1 ) = 1. Let (B3 , T3 ) be the exterior of (B0 , T0 ) in (S 3 , K). Since (B3 , T3 ) is a 2-tangle, we have a tangle decomposition (B0 , T0 ) ∪P1 (B3 , T3 ). Hence we have: (B3 ,T3 )
(S 3 , K) = (B0 , T0 ) ∪ (V1 ∪ V2 ) ∪ (B2 , T2 ) . (B1 ,T1 )
It follows from Corollary 3.3 and Lemma 4.1 that P1 gives an essential 2-tangle decomposition of (S 3 , K) and is not parallel to P2 . Therefore we have the desired example if we take (B2 , T2 ) as an appropriate 2-tangle. Acknowledgements This work was supported by JSPS KAKENHI Grant Number 24740041. The author would like to thank the referee for useful comments. References [1] K. Morimoto, M. Sakuma, Y. Yokota, Examples of tunnel number one knots which have the property “1 + 1 = 3”, Math. Proc. Camb. Philos. Soc. 119 (1) (1996) 113–118. [2] M. Ozawa, On uniqueness of essential tangle decompositions of knots with free tangle decompositions, in: G.T. Jin, K.H. Ko (Eds.), Proc. Appl. Math. Workshop, vol. 8, KAIST, Taejon, 1998, pp. 227–232. [3] T. Saito, Tunnel number of tangles and knots, J. Math. Soc. Jpn. 66 (4) (2014) 1303–1313. [4] M. Tomova, Thin position for knots in a 3-manifold, J. Lond. Math. Soc. 80 (2009) 85–98.