Let M be a compact, connected, orientable and irreducible 3-manifold with a toral boundary component T. The unoriented isotopy class of a non-trivial simple closed curve on T is called its slope, and if γ₁ and γ₂ are slopes on T then Δ (γ₁, γ₂) will denote their minimal geometric intersection number.

Let γ be a slope on T, and let M(γ) be the 3-manifold obtained by γ-Dehn filling. Thus M(γ) = M ∪ J_γ, where J_γ is a solid torus, glued to M along T in such a way that γ bounds a disk in J_γ.

Recall that a surface in a 3-manifold is called essential if it is properly embedded and either (i) incompressible, not parallel to a subsurface of the boundary of the 3-manifold, and not a 2-sphere, or (ii) a 2-sphere that does not bound a 3-ball.

In [6], Thurston has shown that if M is hyperbolic then M(γ) is also hyperbolic, for all but finitely many slopes γ. A slope γ is said to be exceptional if M(γ) is not hyperbolic. Then the exceptional slopes γ has been the subject of many important works in low dimensional topology. This paper is devoted to a special class of exceptional slopes, those which produce a Klein bottle in the resulting 3-manifold. Note that a 3-manifold which contains a Klein bottle contains also an essential 2-torus or an essential 2-sphere, or is Seifert fibered, so is not hyperbolic.

In the case where M is the exterior of a hyperbolic knot in S³, Gordon and Luecke [2] showed that if M(γ) contains a Klein bottle, then Δ (µ, γ)=1. Ichihara and Teragaito [Corollary 1.3, 4] also showed that Δ (λ, γ)≤ 4g, where g is the genus of the knot. It is then natural to ask if it is possible for an arbitrary hyperbolic link K ∪ L, that some Dehn filling on L produces a 3-manifold containing a Klein bottle.

More precisely, let K ∪ L be a hyperbolic link in a homology 3-sphere Y such that their linking number lk(K, L) ≠ 0; we denote the genus of K in Y by g. Denote its exterior by M and let µ be the meridional slope on the boundary of a tubular neighborhood N(L) of L. Let M_L(γ) be the 3-manifold obtained by γ-Dehn filling along ∥ N(L). Then our main result is stated as follows:

Theorem 1.1. If ML(γ) contains a Klein bottle, then

Remark 1.2. Theorem 1.1 can be generalized if we take a link with more components instead the knot K.

Let M be a compact, connected, orientable and irreducible 3-manifold with a toral boundary component T. Let (Fi,∂Fi)⊂(M,∂M) (i=1,2) be an orientable surface, which is possibly compressible or ∥-compressible in M. Suppose that ∂Fi∩T=∅ and that components of ∂F1∩T and of ∂F2∩T have distinct slopes γ₁ and γ₂ on T. We assume that F₁ and F₂ intersect transversely and that ∂F1 and ∂F2 intersect in the minimal number of points so that each component of ∂F1 and each of ∂F2 intersect just Δ(γ1,γ2) times in T. We obtain a surface Fi^ when we cap off the components of ∂Fi∩T with disks.

The intersections of F₁ and F₂ in M give rise to a pair of labeled graphs, G1⊂F1^ and G2⊂F2^. We obtain these graphs as follow: G₁ is the graph obtained by taking as fat vertices the disks F^1− IntF₁ and as edges the arc components of F1∩F2 in F^1. Similarly, G₂ is the graph in F^2 whose vertices are the disks F^2− IntF₂ and whose edges are the arc components of F1∩F2 in F^2. Note that we neglect circle components of F1∩F2. We number the components of ∂F1 1,2,...,n1 in the order in which they appear on T. Similarly, we number the components of ∂F2 1,2,...,n2. This gives a numbering of the vertices of G₁ and G₂. Furthermore, it induces a labelling of the endpoints of edges in G₁ and G₂. On a vertex of G₁ (resp. G₂) one sees the labels 1 through n₂ repeated Δ(γ1,γ2) times (resp. $1$ through $n_1$ repeated Δ(γ1,γ2) times) appearing in order around the vertex. Two vertices on G₁ are parallel if the ordering of the labels on each is clockwise or the ordering on each is anticlockwise, otherwise the vertices are called antiparallel. An edge of G₁ is a positive edge if it connects parallel vertices. Otherwise it is a negative edge. The same applies to vertices of G₂. The orientability of F_i (i =1,2) and M gives us the parity rule: An edge e in G_i is a positive edge if and only if the corresponding edge in G_j is a negative edge.

An edge e is an interior edge if it connects vertices, and a boundary edge otherwise. An edge with labels x and y at its endpoints is called an {x, y}-edge. If an edge has a label x at least one endpoint, it is called an x-edge. An x-cycle is a cycle σ of x-edges of G₁ such that all the vertices of G₁ in σ are parallel and such that the cycle can be oriented so that the tail of each edge has label x. A Scharlemann cycle is an x-cycle that bounds a disk face of G₁ (called a Scharlemann disk). The number of edges in a Scharlemann cycle, σ, is called the length of σ. A pair of adjacent parallel positive edges \{e₁, e₂\} in G₁ is called an S-cycle if it is a Scharlemann cycle of length 2. A Scharlemann cycle is called a trivial loop if its length equals one, and a non-trivial Scharlemann cycle otherwise. Let t_i be the number of trivial loops in the graph G_i, and b_i the number of boundary edges each of which has an end point in a fat vertex. Since b₁ = b₂, we denote this number by b. Then we have the following inequality from Hayashi and Motegi [Theorem 2.1, 3].

Theorem 2.1. Suppos the graph G_i (i=1, 2) contains no non-trivial Scharlemann cycles. Then we have:

As in Section 1, let M be the exterior of a hyperbolic link K∪L in a homology 3-sphere Y, such that the linking number lk(K,L)≠0. Let P=Σ∩M, where Σ is a Seifert surface of minimal genus g ≥ 1 for the knot K. Now, among all these minimal genus surfaces, we choose P such that p=|∂P∩∂N(L)| is minimal. Then P is incompressible and boundary-incompressible in M. Note that p≥|lk(K,L)|. Let ML(γ) be the 3-manifold obtained by γ-Dehn filling along ∂N(L). Suppose that ML(γ) contains a Klein bottle S^ such that s=|S^∩Jγ| is minimal and s > 1. Let S=S^∩M, then S is incompressible and boundary-incompressible in M.

Let T^=∂N(S^), where N(S^) denotes a thin regular neighborhood of S^ in M(γ). Then T^ is a 2-torus, intersecting J_γ in t=2s points. Let T=T^∩M.

As above, the intersection P∩T gives rise to a pair of labeled graphs, GP⊂P^⊂ML(μ) and GT⊂T^⊂ML(γ). The choice of P^, S^ and N(S^) guarantees that the intersection P∩T is essential: no circle component of P∩T bounds a disk in P or in T, and no arc component of P∩T is boundary-parallel in either P or T. In particular neither G_P nor G_T contains trivial loops. Also, since T∩∂N(K)=∅, then the graphs G_P and G_T have no boundary edges. Since the torus T^ is separating, then T^ divides ML(γ) in a black side, U, and a white side W. Therefore, the faces of G_P are divided into black and white faces. Furthermore each black face of G_P is a face of length 2, corresponding to the regular neighborhood of an arc component of P∩S. See [2] (pages 626-628) for more details.

Lemma 3.1. s is odd.

Proof. If s is even, then we can obtain a closed non-orientable surface in M by attaching suitable annuli in ∂M to S along ∂S, which impossible (since the homology sphere Y does not contain a closed non-orientable surface).

Lemma 3.2. G_T does not contain a Scharlemann cycle.

Proof. Assume for contradiction that G_T contains a Scharlemann cycle. Now by a standard argument of surgery (see for example [Lemma 2.5.2, 1]), we can construct a new surface P' of genus g such that ∂0P′=∂0P and |∂P′∩∂N(L)|=p−2 contradicting the minimality of p.

Now let {σj}j=1,…,m denote the set of all Scharlemann cycles in G_P and let {Dj}j=1,…,m be the set of their corresponding Scharlemann disks in G_P.

The following Lemma can be proved by using [Lemma 2.1, 5]. However for convenience of the readers we give a proof here.

Lemma 3.3. m≥(Δ(μ,γ)−2)p−2g+1

Proof. Let P^×[−ϵ,ϵ] be a small regular neighborhood of P^, and let R^ be the closed surface of genus 2g obtained by taking two copies of P^ glued together along a small regular neighborhood C of ∂0P. R^=(P^×{−ϵ})∪C∪(P^×{ϵ}). Without loss of generality we can assume that R^ is contained in Y−∂N(K) (we push slightly R^ in Y−∂N(K)). Let R=R^∩M, then ∂R is a collection of 2p components, each having the meridional slope µ in ∂N(L). As above, we define the labelled graphs of intersection, G_R in R^ and G_T in T^. We note that G_R is made of two disjoint copies of G_P.

By hypothesis the graph G_P contains only m Scharlemann cycles, so G_R contains exactly 2m Scharlemann cycles. Recall that for each Scharlemann cycle σ_j in G_P there exists a disk face D_j such that GP∩intDj=∅. For each j, let B_j a small disk contained in intD_j. Now, we consider the annuli H_j in Bj×[−ϵ,ϵ], such that one boundary component of H_j lies in Bj×{−ϵ}, while the other lies in Bj×{ϵ}. Let H=H1∪H2∪…∪Hm. Let F^ be the resulting surface obtained by deleting B_j's and then attaching H to R^. It is easy to see that F^ is a closed surface of genus 2g+m. As usual, let F=F^∩M. Let (G_F, G_T) be the pair of intersection graphs associated to the pair of surfaces (F, T). Note that G_F has no Scharlemann cycle.

Now by Hayashi-Motegi's inequality [Theorem 2.1, 3], applied to the two graphs obtained from F∩T we get:

Hence, the proof is completed.

Lemma 3.4. If σ , τ and κ are Scharlemann cycles in G_P, then two of them must have the same pair of labels.

Proof. See [Theorem 6.5, 2].

Lemma 3.5. If t ≥ 4 then G_P does not contain a white Scharlemann disk. In particular every Scharlemann disk is a face of length 2.

Proof. See [Theorem 6.4, 2].

Proof of Theorem 1.1. Suppose that G_P contains m Scharlemann cycles. By Lemma 3.5, any Scharlemann cycle in G_P bounds a black face, hence any Scharlemann cycle σ in G_P is a black S-cycle. Let {σj1}j=1,…,m1 denote the set of all black S-cycles in G_P having {x, x+1} as labels, and let {σj2}j=1,…,m2 be the set of all black S-cycles in G_P having {y, y+1} as labels such that {x,x+1}≠{y,y+1}. Since the torus T^ is separating, then x and y must have the same parity.

By Lemma 3.4, m₁+m₂=m. Recall that for each S-cycle σj1 (resp. σj2) in G_P there exists a black disk face Dj1 (resp. a black disk face Dj2) such that GP∩intDj1=∅ (resp. GP∩intDj2=∅).

Without loss of generality, we assume that m1≥m2 and {x,x+1}={1,2}. Now let Γ be the subgraph of G_P consisting of the vertices of G_P and all the edges of all the black S-cycles in G_P having {1, 2} as labels. Denote by f_k the faces of Γ. Now an Euler characteristic count gives

Now let F be the number of disk faces of Γ which are not a Scharlemann disk {Dj1}j=1,…,m1. Then

Lemma 4.1. Γ contains exactly m₁ disks faces.

Proof. We argue by contradiction. Suppose that Γ contains a cycle σ bounding a disk face E, such that E is not a black Scharlemann disk. By construction of Γ, all the vertices of σ are parallel and {1, 2} are the labels of σ.

Now, let e be an edge of GP∩E. e is called a diagonal edge if e is incident to a vertex v and has other end in a vertex w which is not adjacent to v on σ.

We divide in two cases depending upon whether a diagonal edge e exists or not.

Case 1: There is no diagonal edge in GP∩E.

Let e be an edge of GP∩E. Therefore, if k is the label at one endpoint of e then 2s+3-k (mod 2s) is the label at the other endpoint

In particular, if e is an (s+1)-edge or an (s+2)-edge, then e is an {s+1, s+2}-edge. Note that since s > 1, then s+1,s+2={1,2}..

Now, let e and f be two adjacent {s+1, s+2}-edges of GP∩E. If e and f are parallel, then {e, f} is an S-cycle with {s+1, s+2} as labels. Since s+1 is even and {1, 2} are the labels of all the black S-cycles of Γ, then {e, f} is a white S-cycle, which contradicts Lemma 3.5. If not, G_P must contain a white Scharlemann cycle with {s+1, s+2} as labels and having the same length as σ, but this is impossible by Lemma 3.5.

Case 2: There is a diagonal edge e in GP∩E.

Let e be a diagonal edge connecting two vertices v and w of σ. Let i be the label at the end of e at v and j be the label at the end of e at w. The diagonal edge e divides the disk E into two disks E₁ and E₂. Without loss of generality, we may assume that there is no other diagonal edge in GP∩E1. Let σ' be the cycle bounding E₁.

As in Case 1, if two adjacent {s+1, s+2}-edges of GP∩E1 are parallel, then G_P must contain a white S-cycle, which contradicts Lemma 3.5. If not, let u be a vertex of σ' adjacent to the vertex v and u≠w, then only one {s+1, s+2}-edge of GP∩E1 connect u to v, hence i∈s+1,s+2. Using the same argument we show that j∈{s+1,s+2}. Since v and w are parallel, then by the parity rule i must be different to j. Hence {i,j}={s+1,s+2} and e must be an {s+1, s+2}-edge. Then G_P must contain a white Scharlemann cycle with {s+1, s+2} as labels and having the same length as σ', but this also impossible by Lemma 3.5.

By the previous lemma, F=0. Hence we obtain:

Since m1≥m2, this yields

Now by Lemma 3.3, we get

Hence, the proof of Theorem 1.1 is completed.

We would like to thank the editor and reviewers for careful reading, and constructive suggestions for our manuscript. We would like to thank Professor Rebhi Salem for helpful comments.