Abstract.
We find a self-linking number formula for a given null-homologous transverse link in a contact manifold that is compatible with either an annulus or a pair of pants open book decomposition. It extends Bennequin’s self-linking formula for a braid in the standard contact 3333-sphere.

2000 Mathematics Subject Classification:
Primary 57M25, 57M27; Secondary 57M50

The first author was partially supported by NSF grants DMS-0806492 and DMS-0635607.

Alexander’s theorem [1] states that every closed and oriented 3333-manifold admits an open book decomposition.

Definition 1.1.

Let ΣΣ\Sigmaroman_Σ be a surface with non empty boundary and ϕitalic-ϕ\phiitalic_ϕ be a diffeomorphism of the surface fixing the boundary pointwise. We construct a closed manifold

M(Σ,ϕ)=Σ×[0,1]/∼fragmentssubscriptݑ€Σitalic-ϕΣfragments[0,1]similar-toM_(\Sigma,\phi)=\Sigma\times[0,1]/\simitalic_M start_POSTSUBSCRIPT ( roman_Σ , italic_ϕ ) end_POSTSUBSCRIPT = roman_Σ × [ 0 , 1 ] / ∼
where “∼similar-to\sim∼” is an equivalence relation satisfying (ϕ(x),0)∼(x,1)similar-toitalic-ϕݑ¥0ݑ¥1(\phi(x),0)\sim(x,1)( italic_ϕ ( italic_x ) , 0 ) ∼ ( italic_x , 1 ) for x∈Int(Σ)ݑ¥IntΣx\in\rm Int(\Sigma)italic_x ∈ roman_Int ( roman_Σ ) and (x,τ)∼(x,1)similar-toݑ¥ݜݑ¥1(x,\tau)\sim(x,1)( italic_x , italic_τ ) ∼ ( italic_x , 1 ) for x∈∂Σݑ¥Σx\in\partial\Sigmaitalic_x ∈ ∂ roman_Σ and τ∈[0,1]ݜ01\tau\in[0,1]italic_τ ∈ [ 0 , 1 ]. The pair (Σ,ϕ)Σitalic-ϕ(\Sigma,\phi)( roman_Σ , italic_ϕ ) is called an abstract open book decomposition of the manifold M(Σ,ϕ)subscriptݑ€Σitalic-ϕM_(\Sigma,\phi)italic_M start_POSTSUBSCRIPT ( roman_Σ , italic_ϕ ) end_POSTSUBSCRIPT.

Alternatively, an open book decomposition for Mݑ€Mitalic_M can be defined as a pair (L, πݜ‹\piitalic_π), where (1) Lݐ¿Litalic_L is an oriented link in Mݑ€Mitalic_M called the binding of the open book; (2) π:M∖L→S1:ݜ‹→ݑ€ݐ¿superscriptݑ†1\pi:M\setminus L\to S^1italic_π : italic_M ∖ italic_L → italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT is a fibration whose fiber, π-1(θ)superscriptݜ‹1ݜƒ\pi^-1(\theta)italic_π start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_θ ), called a page, is the interior of a compact surface Σθ⊂MsubscriptΣݜƒݑ€\Sigma_\theta\subset Mroman_Σ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ⊂ italic_M such that ∂Σθ=LsubscriptΣݜƒݐ¿\partial\Sigma_\theta=L∂ roman_Σ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT = italic_L for all θ∈S1ݜƒsuperscriptݑ†1\theta\in S^1italic_θ ∈ italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT.

One of the central results about the topology of contact 3333-manifolds is Giroux correspondence [9]:

contact structures ξ on M3up to contact isotopy⟷1-1open book decompositions (Σ,ϕ)of M3 up to positive stabilization.superscript⟷11contact structures ξ on M3up to contact isotopyopen book decompositions (Σ,ϕ)of M3 up to positive stabilization\left\\beginarray[]l\mboxcontact structures $\xi$ on $M^3$\\ \mboxup to contact isotopy\endarray\right\\stackrel\scriptstyle 1-1% \longleftrightarrow\left\\beginarray[]l\mboxopen book decompositions $% (\Sigma,\phi)$\\ \mboxof $M^3$ up to positive stabilization\endarray\right\. start_ARRAY start_ROW start_CELL contact structures italic_ξ on italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL up to contact isotopy end_CELL end_ROW end_ARRAY start_RELOP SUPERSCRIPTOP start_ARG ⟷ end_ARG start_ARG 1 - 1 end_ARG end_RELOP start_ARRAY start_ROW start_CELL open book decompositions ( roman_Σ , italic_ϕ ) end_CELL end_ROW start_ROW start_CELL of italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT up to positive stabilization end_CELL end_ROW end_ARRAY .
For example, the standard contact structure ξstd=ker(dz+r2dθ)subscriptݜ‰ݑ ݑ¡ݑ‘kernelݑ‘ݑ§superscriptݑŸ2ݑ‘ݜƒ\xi_std=\ker(dz+r^2d\theta)italic_ξ start_POSTSUBSCRIPT italic_s italic_t italic_d end_POSTSUBSCRIPT = roman_ker ( italic_d italic_z + italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d italic_θ ) on S3=ℝ3∪{∞}superscriptݑ†3superscriptℝ3S^3=\mathbbR^3\cup\\infty\italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT = blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ∪ { ∞ } corresponds to the open book decomposition (D2,id)superscriptݐ·2ݑ–ݑ‘(D^2,id)( italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_i italic_d ).

We define a braid and the braid index in a general open book setting:

Definition 1.2.

Suppose (L,π)ݐ¿ݜ‹(L,\pi)( italic_L , italic_π ) is an open book decomposition for a 3333-manifold Mݑ€Mitalic_M. A link K⊂Mݐ¾ݑ€K\subset Mitalic_K ⊂ italic_M is called a (closed) braid if Kݐ¾Kitalic_K transversely intersects each page Σθ=π-1(θ)subscriptΣݜƒsuperscriptݜ‹1ݜƒ\Sigma_\theta=\pi^-1(\theta)roman_Σ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT = italic_π start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_θ ) of the open book. That is, at each point p∈K∩Σθݑݐ¾subscriptΣݜƒp\in K\cap\Sigma_\thetaitalic_p ∈ italic_K ∩ roman_Σ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT, we have TpΣθ⊕TpK=TpMdirect-sumsubscriptݑ‡ݑsubscriptΣݜƒsubscriptݑ‡ݑݐ¾subscriptݑ‡ݑݑ€T_p\Sigma_\theta\oplus T_pK=T_pMitalic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT roman_Σ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ⊕ italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_K = italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_M. The braid index of a braid Kݐ¾Kitalic_K is the degree of the map πݜ‹\piitalic_π restricted to Kݐ¾Kitalic_K. In other words, if a braid Kݐ¾Kitalic_K intersects each page in nݑ›nitalic_n points, then the braid index of Kݐ¾Kitalic_K is nݑ›nitalic_n.

Bennequin [2] proved that any transverse link in (S3,ξstd)superscriptݑ†3subscriptݜ‰ݑ ݑ¡ݑ‘(S^3,\xi_std)( italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_s italic_t italic_d end_POSTSUBSCRIPT ) can be transversely isotoped to a closed braid in (D2,id)superscriptݐ·2ݑ–ݑ‘(D^2,id)( italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_i italic_d ). Later the second author generalized Bennequin’s result into the following:

Theorem 1.3.

[13, Theorem 3.2.1] Suppose (Σ,ϕ)normal-Σitalic-ϕ(\Sigma,\phi)( roman_Σ , italic_ϕ ) is an open book decomposition for a 3333-manifold M=M(Σ,ϕ)ݑ€subscriptݑ€normal-Σitalic-ϕM=M_(\Sigma,\phi)italic_M = italic_M start_POSTSUBSCRIPT ( roman_Σ , italic_ϕ ) end_POSTSUBSCRIPT. Let ξ=ξ(Σ,ϕ)ݜ‰subscriptݜ‰normal-Σitalic-ϕ\xi=\xi_(\Sigma,\phi)italic_ξ = italic_ξ start_POSTSUBSCRIPT ( roman_Σ , italic_ϕ ) end_POSTSUBSCRIPT be a compatible contact structure. Let Kݐ¾Kitalic_K be a transverse link in (M,ξ)ݑ€ݜ‰(M,\xi)( italic_M , italic_ξ ). Then Kݐ¾Kitalic_K can be transversely isotoped to a braid in (Σ,ϕ)normal-Σitalic-ϕ(\Sigma,\phi)( roman_Σ , italic_ϕ ).

The self linking (normal-(((Bennequin)normal-))) number is a classical invariant for transverse knots. Bennequin [2] gave a formula of the self linking number for a braid bݑbitalic_b in (D2,id)superscriptݐ·2ݑ–ݑ‘(D^2,id)( italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_i italic_d ):

(1.1) sl(b)=-n+a,ݑ ݑ™ݑݑ›ݑŽsl(b)=-n+a,italic_s italic_l ( italic_b ) = - italic_n + italic_a ,
where nݑ›nitalic_n is the braid index, and aݑŽaitalic_a the algebraic crossing number (the exponent sum) of the braid.

The first goal of this paper is to give a combinatorial page_content for the self linking number of a null-homologous transverse link in the contact lens spaces compatible with (A,Dk)ݐ´superscriptݐ·ݑ˜(A,D^k)( italic_A , italic_D start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) the annulus Aݐ´Aitalic_A open book decomposition with monodromy the kthsuperscriptݑ˜ݑ¡ℎk^thitalic_k start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT power of the positive Dehn twist Dݐ·Ditalic_D. By Theorem 1.3, our problem is reduced to searching a self linking formula for a null-homologous braid in the open book decomposition (A,Dk)ݐ´superscriptݐ·ݑ˜(A,D^k)( italic_A , italic_D start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ). Such a braid is given by a product of permutations of points in a local disk on the annulus Aݐ´Aitalic_A and moves of points which turn around the hole of Aݐ´Aitalic_A. We denote by aσsubscriptݑŽݜŽa_\sigmaitalic_a start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT the algebraic crossing number of the local permutations, and by aρsubscriptݑŽݜŒa_\rhoitalic_a start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT the algebraic rotation number around the hole of Aݐ´Aitalic_A, see Definition 2.5 for precise definitions. With these notations, we extend Bennequin’s formula (1.1) into the following:

Theorem 4.1. Let bݑbitalic_b be a null-homologous closed braid in (A,Dk)ݐ´superscriptݐ·ݑ˜(A,D^k)( italic_A , italic_D start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) of braid index nݑ›nitalic_n. For k≠0ݑ˜0k\neq 0italic_k ≠0 we have

sl(b)=-n+aσ+aρ(1-aρk).ݑ ݑ™ݑݑ›subscriptݑŽݜŽsubscriptݑŽݜŒ1subscriptݑŽݜŒݑ˜sl(b)=-n+a_\sigma+a_\rho(1-\fraca_\rhok).italic_s italic_l ( italic_b ) = - italic_n + italic_a start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT ( 1 - divide start_ARG italic_a start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG italic_k end_ARG ) .
When k=0ݑ˜0k=0italic_k = 0 there exists a canonical Seifert surface Σbsubscriptnormal-Σݑ\Sigma_broman_Σ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT of bݑbitalic_b and we have

sl(b,Σb)=-n+aσ.ݑ ݑ™ݑsubscriptΣݑݑ›subscriptݑŽݜŽsl(b,\Sigma_b)=-n+a_\sigma.italic_s italic_l ( italic_b , roman_Σ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ) = - italic_n + italic_a start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT .

The Seifert surface ΣbsubscriptΣݑ\Sigma_broman_Σ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT will be constructed in Section 3. The surface is canonical in the sense that the way of construction is similar to that of the standard Seifert surface, or Bennequin surface, of a closed braid in S3superscriptݑ†3S^3italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT.

Our second goal is to find a self-linking formula for null-homologous transverse links in a contact Seifert fibered manifold Mݑ€Mitalic_M of signature (g=0,k1,k2,k3)ݑ”0subscriptݑ˜1subscriptݑ˜2subscriptݑ˜3(g=0,k_1,k_2,k_3)( italic_g = 0 , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ). Let Sݑ†Sitalic_S be a pair of pants (a disk with two holes). Let Disubscriptݐ·ݑ–D_iitalic_D start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT (i=1,2,3ݑ–123i=1,2,3italic_i = 1 , 2 , 3) be the positive Dehn twists along the curves parallel to the boundary circles of Sݑ†Sitalic_S. Then Mݑ€Mitalic_M has an open book decomposition (S,D1k1∘D2k2∘D3k3)ݑ†superscriptsubscriptݐ·1subscriptݑ˜1superscriptsubscriptݐ·2subscriptݑ˜2superscriptsubscriptݐ·3subscriptݑ˜3(S,D_1^k_1\circ D_2^k_2\circ D_3^k_3)( italic_S , italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∘ italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∘ italic_D start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ), and is equipped with a compatible contact structure. A braid in the pants open book is a product of permutations of points in a local disk on Sݑ†Sitalic_S and moves of points which turn around the holes of Sݑ†Sitalic_S. We denote by aσsubscriptݑŽݜŽa_\sigmaitalic_a start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT the algebraic crossing number of the local permutations and by aρisubscriptݑŽsubscriptݜŒݑ–a_\rho_iitalic_a start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT (i=2,3ݑ–23i=2,3italic_i = 2 , 3) the algebraic winding number around the holes. See Definition 5.4 for precise definitions. We obtain the following formula which also extends (1.1).

Theorem 5.6 Let bݑbitalic_b be a null-homologous braid in (S,D1k1∘D2k2∘D3k3)ݑ†superscriptsubscriptݐ·1subscriptݑ˜1superscriptsubscriptݐ·2subscriptݑ˜2superscriptsubscriptݐ·3subscriptݑ˜3(S,D_1^k_1\circ D_2^k_2\circ D_3^k_3)( italic_S , italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∘ italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∘ italic_D start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) of braid index nݑ›nitalic_n. Suppose k1,k2,k3subscriptݑ˜1subscriptݑ˜2subscriptݑ˜3k_1,k_2,k_3italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT are integers with k1,k2,k3≥0subscriptݑ˜1subscriptݑ˜2subscriptݑ˜30k_1,k_2,k_3\geq 0italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ≥ 0; k1,k2,k3≤0subscriptݑ˜1subscriptݑ˜2subscriptݑ˜30k_1,k_2,k_3\leq 0italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ≤ 0; or k1=0,k2k3<0formulae-sequencesubscriptݑ˜10subscriptݑ˜2subscriptݑ˜30k_1=0,k_2k_3<0italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 , italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT <0. We have:

sl(b,[Σb])=-n+aσ+aρ2(1-s2)+aρ3(1-s3)-(s2+s3)k1,ݑ ݑ™ݑdelimited-[]subscriptΣݑݑ›subscriptݑŽݜŽsubscriptݑŽsubscriptݜŒ21subscriptݑ 2subscriptݑŽsubscriptݜŒ31subscriptݑ 3subscriptݑ 2subscriptݑ 3subscriptݑ˜1sl(b,[\Sigma_b])=-n+a_\sigma+a_\rho_2(1-s_2)+a_\rho_3(1-s_3)-(% s_2+s_3)k_1,italic_s italic_l ( italic_b , [ roman_Σ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ] ) = - italic_n + italic_a start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT + italic_a start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( 1 - italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_a start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( 1 - italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) - ( italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ,
where Σbsubscriptnormal-Σݑ\Sigma_broman_Σ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT is some Seifert surface for bݑbitalic_b. The constants s2subscriptݑ 2s_2italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, s3subscriptݑ 3s_3italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT are determined by aρ2subscriptݑŽsubscriptݜŒ2a_\rho_2italic_a start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, aρ3subscriptݑŽsubscriptݜŒ3a_\rho_3italic_a start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, k1subscriptݑ˜1k_1italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, k2subscriptݑ˜2k_2italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and k3subscriptݑ˜3k_3italic_k start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, under the assumption that bݑbitalic_b is null-homologous, see Definition 5.4.

The organization of the paper is the following:

In Section 2, we fix notations and study properties of the contact lens space (M(A,Dk),ξk)subscriptݑ€ݐ´superscriptݐ·ݑ˜subscriptݜ‰ݑ˜(M_(A,D^k),\xi_k)( italic_M start_POSTSUBSCRIPT ( italic_A , italic_D start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ).

In Section 3, we construct a Bennequin type Seifert surface F^bsubscript^ݐ¹ݑ\hatF_bover^ start_ARG italic_F end_ARG start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT for a given braid bݑbitalic_b in (A,Dk)ݐ´superscriptݐ·ݑ˜(A,D^k)( italic_A , italic_D start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ). In general, this F^bsubscript^ݐ¹ݑ\hatF_bover^ start_ARG italic_F end_ARG start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT is an immersed surface and the Bennequin-Eliashberg inequality is not satisfied even for tight cases. We resolve all the singularities and obtain an embedded surface ΣbsubscriptΣݑ\Sigma_broman_Σ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT. We develop a theory about resolution of singularities of an immersed surface and corresponding changes in characteristic foliations.

In Section 4, we prove Theorem 4.1, an explicit formula of the self linking number relative to ΣbsubscriptΣݑ\Sigma_broman_Σ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT, which extends Bennequin’s formula (1.1). As the self linking number is defined to be the euler number of the contact 2222-plane bundle relative to the surface framing, we measure the difference between the immersed F^bsubscript^ݐ¹ݑ\hatF_bover^ start_ARG italic_F end_ARG start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT-framing and the embedded ΣbsubscriptΣݑ\Sigma_broman_Σ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT-framing. We also study the behavior of our self linking number under a braid stabilization. Corollary 4.5 states that our self linking number is invariant under a positive stabilization and changes by 2222 under a negative stabilization, which extends Bennequin’s result for braids in (S3,ξstd)superscriptݑ†3subscriptݜ‰ݑ ݑ¡ݑ‘(S^3,\xi_std)( italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_s italic_t italic_d end_POSTSUBSCRIPT ).

In Section 5, we apply our surface construction method to some class of contact Seifert fibered manifolds and prove Theorem 5.6.

Acknowledgements. The authors would like to thank John Etnyre for numerous useful comments and sharing his ideas, especially those on Corollary 3.9, and Matthew Hedden for helpful comments on Section 4. They also thank the referee for carefully examining the paper and providing constructive comments. K.K. thanks Tim Cochran and Walter Neumann for stimulus conversations.

Let A=S1×Iݐ´superscriptݑ†1ݐ¼A=S^1\times Iitalic_A = italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT × italic_I be an annulus and Dαsubscriptݐ·ݛ¼D_\alphaitalic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT the positive Dehn twist about the core circle α=S1×12ݛ¼superscriptݑ†112\alpha=S^1\times\\frac12\italic_α = italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT × divide start_ARG 1 end_ARG start_ARG 2 end_ARG . For simplicity, we denote Dαsubscriptݐ·ݛ¼D_\alphaitalic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT by Dݐ·Ditalic_D.

We study an abstract open book decomposition (A,Dk)ݐ´superscriptݐ·ݑ˜(A,D^k)( italic_A , italic_D start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ).

Claim 2.1.

The corresponding manifold M(A,Dk)subscriptݑ€ݐ´superscriptݐ·ݑ˜M_(A,D^k)italic_M start_POSTSUBSCRIPT ( italic_A , italic_D start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT to (A,Dk)ݐ´superscriptݐ·ݑ˜(A,D^k)( italic_A , italic_D start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) is:

M(A,Dk)={L(k,k-1) if k>0,S1×S2 if k=0,L(|k|,1) if k<0.subscriptݑ€ݐ´superscriptݐ·ݑ˜casesݐ¿ݑ˜ݑ˜1 if ݑ˜0superscriptݑ†1superscriptݑ†2 if ݑ˜0ݐ¿ݑ˜1 if ݑ˜0M_{(A,D^{k})}=\left\{\begin{array}[]{ll}L(k,k-1)&\mbox{ if }k>0,\\ S^{1}\times S^{2}&\mbox{ if }k=0,\\ L(|k|,1)&\mbox{ if }k<0.\end{array}\right.italic_M start_POSTSUBSCRIPT ( italic_A , italic_D start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT = { start_ARRAY start_ROW start_CELL italic_L ( italic_k , italic_k - 1 ) end_CELL start_CELL if italic_k >0 , end_CELL end_ROW start_ROW start_CELL italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT × italic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL if italic_k = 0 , end_CELL end_ROW start_ROW start_CELL italic_L ( | italic_k | , 1 ) end_CELL start_CELL if italic_k <0 . end_CELL end_ROW end_ARRAY

Let D∘≃D2similar-to-or-equalssubscriptݐ·superscriptݐ·2D_{\circ}\simeq D^{2}italic_D start_POSTSUBSCRIPT ∘ end_POSTSUBSCRIPT ≃ italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT be a disk and γ:=∂D∘assignݛ¾subscriptݐ·\gamma:=\partial D_{\circ}italic_γ := ∂ italic_D start_POSTSUBSCRIPT ∘ end_POSTSUBSCRIPT. Recall that (D∘,id)subscriptݐ·ݑ–ݑ‘(D_{\circ},id)( italic_D start_POSTSUBSCRIPT ∘ end_POSTSUBSCRIPT , italic_i italic_d ) is a planar open book decomposition for (S3,ξstd)superscriptݑ†3subscriptݜ‰ݑ ݑ¡ݑ‘(S^{3},\xi_{std})( italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_s italic_t italic_d end_POSTSUBSCRIPT ). Let Dμ⊂D∘subscriptݐ·ݜ‡subscriptݐ·D_{\mu}\subset D_{\circ}italic_D start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ⊂ italic_D start_POSTSUBSCRIPT ∘ end_POSTSUBSCRIPT be a disc with boundary μݜ‡\muitalic_μ. The core of the solid torus Dμ×S1⊂S3subscriptݐ·ݜ‡superscriptݑ†1superscriptݑ†3D_{\mu}\times S^{1}\subset S^{3}italic_D start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT × italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ⊂ italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT is the unknot, UݑˆUitalic_U. The meridian of the torus Tμ=∂(Dμ×S1)subscriptݑ‡ݜ‡subscriptݐ·ݜ‡superscriptݑ†1T_{\mu}=\partial(D_{\mu}\times S^{1})italic_T start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = ∂ ( italic_D start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT × italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) is μݜ‡\muitalic_μ. Pick a point p∈μݑݜ‡p\in\muitalic_p ∈ italic_μ, and define a longitude λݜ†\lambdaitalic_λ of Tμsubscriptݑ‡ݜ‡T_{\mu}italic_T start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT as λ={p}×S1ݜ†ݑsuperscriptݑ†1\lambda=\{p\}\times S^{1}italic_λ = { italic_p } × italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT. Remove Dμ×S1subscriptݐ·ݜ‡superscriptݑ†1D_{\mu}\times S^{1}italic_D start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT × italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT from S3superscriptݑ†3S^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT, and attach a new solid torus by identifying its meridian mݑšmitalic_m with λݜ†\lambdaitalic_λ and its longitude lݑ™litalic_l with -μݜ‡-\mu- italic_μ. This is the 00-surgery along the unknot UݑˆUitalic_U. The resulting manifold is S1×S2superscriptݑ†1superscriptݑ†2S^{1}\times S^{2}italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT × italic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. In this way we get an open book decomposition (A,idA)ݐ´ݑ–subscriptݑ‘ݐ´(A,id_{A})( italic_A , italic_i italic_d start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) for S1×S2superscriptݑ†1superscriptݑ†2S^{1}\times S^{2}italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT × italic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, whose page Aݐ´Aitalic_A is the union of the annulus D∘∖Dμsubscriptݐ·subscriptݐ·ݜ‡D_{\circ}\setminus D_{\mu}italic_D start_POSTSUBSCRIPT ∘ end_POSTSUBSCRIPT ∖ italic_D start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT, shaded in Figure 2-(1), and the annulus bounded by -lݑ™-l- italic_l and the core γ′superscriptݛ¾′\gamma^{\prime}italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT of the solid torus, sketched in Figure 2-(2).

The Dehn twist Dksuperscriptݐ·ݑ˜D^{k}italic_D start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT about the core U′⊂(D∘∖Dμ)⊂Asuperscriptݑˆ′subscriptݐ·subscriptݐ·ݜ‡ݐ´U^{\prime}\subset(D_{\circ}\setminus D_{\mu})\subset Aitalic_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊂ ( italic_D start_POSTSUBSCRIPT ∘ end_POSTSUBSCRIPT ∖ italic_D start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ) ⊂ italic_A, sketched in Figure 2-(3), of the page annulus Aݐ´Aitalic_A is equivalent to applying (1-k)1ݑ˜(\frac{1}{-k})( divide start_ARG 1 end_ARG start_ARG - italic_k end_ARG )-surgery along the unknot U′superscriptݑˆ′U^{\prime}italic_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. The link (U∪U′)⊂S3ݑˆsuperscriptݑˆ′superscriptݑ†3(U\cup U^{\prime})\subset S^{3}( italic_U ∪ italic_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ⊂ italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT is the positive Hopf link. By the slam-dunk operation, the surgery page_content is reduced to the kݑ˜kitalic_k-surgery along UݑˆUitalic_U, which represents L(k,-1)=L(k,k-1)ݐ¿ݑ˜1ݐ¿ݑ˜ݑ˜1L(k,-1)=L(k,k-1)italic_L ( italic_k , - 1 ) = italic_L ( italic_k , italic_k - 1 ) when k>0ݑ˜0k>0italic_k >0 and L(|k|,1)ݐ¿ݑ˜1L(|k|,1)italic_L ( | italic_k | , 1 ) when k<0ݑ˜0k<0italic_k <0. ∎

Let (M(A,Dk),ξk)subscriptݑ€ݐ´superscriptݐ·ݑ˜subscriptݜ‰ݑ˜(M_{(A,D^{k})},\xi_{k})( italic_M start_POSTSUBSCRIPT ( italic_A , italic_D start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) be the contact manifold corresponds to the open book (A,Dk)ݐ´superscriptݐ·ݑ˜(A,D^{k})( italic_A , italic_D start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ).

Claim 2.2.

The contact manifold (M(A,Dk),ξk)subscriptݑ€ݐ´superscriptݐ·ݑ˜subscriptݜ‰ݑ˜(M_{(A,D^{k})},\xi_{k})( italic_M start_POSTSUBSCRIPT ( italic_A , italic_D start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) is overtwisted if and only if k<0ݑ˜0k<0italic_k <0. When k≥0ݑ˜0k\geq 0italic_k ≥ 0, this ξksubscriptݜ‰ݑ˜\xi_{k}italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is the unique tight contact structure for L(k,k-1)ݐ¿ݑ˜ݑ˜1L(k,k-1)italic_L ( italic_k , italic_k - 1 ).

If k<0ݑ˜0k<0italic_k <0, Goodman’s criterion for overtwistedness [10, Theorem 1.2] implies that ξksubscriptݜ‰ݑ˜\xi_{k}italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is overtwisted.

When k=0ݑ˜0k=0italic_k = 0, according to [6, proof of Lemma 3.2] of Etnyre-Honda, the open book is a boundary of a positive Lefschetz fibration on a 4444-manifold Xݑ‹Xitalic_X, so that (S1×S2,ξ0)superscriptݑ†1superscriptݑ†2subscriptݜ‰0(S^{1}\times S^{2},\xi_{0})( italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT × italic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_ξ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) is Stein filled by Xݑ‹Xitalic_X, hence tight. Moreover, ξ0subscriptݜ‰0\xi_{0}italic_ξ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the unique tight contact structure on S2×S1superscriptݑ†2superscriptݑ†1S^{2}\times S^{1}italic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT × italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT due to Eliashberg [3].

When k>0ݑ˜0k>0italic_k >0, the monodromy is a product of positive Dehn twists. Etnyre-Honda’s [6, Lemma 3.2] guarantees that the contact structure compatible with such an open book is Stein fillable, hence tight. The uniqueness for k>0ݑ˜0k>0italic_k >0 follows from Honda’s classification of tight contact structures for lens spaces [11]. More precisely, we have

-kk-1=-2-1-2-1-2-⋯-1-2=[-2,-2,⋯,-2], repeating (k-1)-timesformulae-sequenceݑ˜ݑ˜121212⋯1222⋯2 repeating (k-1)-times-\frac{k}{k-1}=-2-\frac{1}{-2-\frac{1}{-2-\cdots-\frac{1}{-2}}}=[-2,-2,\cdots,% -2],\mbox{ \small repeating $(k-1)$-times}- divide start_ARG italic_k end_ARG start_ARG italic_k - 1 end_ARG = - 2 - divide start_ARG 1 end_ARG start_ARG - 2 - divide start_ARG 1 end_ARG start_ARG - 2 - ⋯ - divide start_ARG 1 end_ARG start_ARG - 2 end_ARG end_ARG end_ARG = [ - 2 , - 2 , ⋯ , - 2 ] , repeating (k-1)-times
and |(-2+1)(-2+1)⋯(-2+1)|=12121⋯211|(-2+1)(-2+1)\cdots(-2+1)|=1| ( - 2 + 1 ) ( - 2 + 1 ) ⋯ ( - 2 + 1 ) | = 1, thus the manifold has the unique tight contact structure. ∎

We fix notations. See Figure 3. Suppose we have a null-homologous closed braid bݑbitalic_b of braid index nݑ›nitalic_n in the open book (A,Dk)ݐ´superscriptݐ·ݑ˜(A,D^{k})( italic_A , italic_D start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ). Let γ∪γ′=∂Aݛ¾superscriptݛ¾′ݐ´\gamma\cup\gamma^{\prime}=\partial Aitalic_γ ∪ italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ∂ italic_A whose orientations are induced by that of Aݐ´Aitalic_A. Let Aθsubscriptݐ´ݜƒA_{\theta}italic_A start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT (θ∈[0,1]ݜƒ01\theta\in[0,1]italic_θ ∈ [ 0 , 1 ]) denote the page A×{θ}⊂M(A,Dk)ݐ´ݜƒsubscriptݑ€ݐ´superscriptݐ·ݑ˜A\times\{\theta\}\subset M_{(A,D^{k})}italic_A × { italic_θ } ⊂ italic_M start_POSTSUBSCRIPT ( italic_A , italic_D start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT. Under the identification A=S1×[0,1]ݐ´superscriptݑ†101A=S^{1}\times[0,1]italic_A = italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT × [ 0 , 1 ], we set α=S1×{12}ݛ¼superscriptݑ†112\alpha=S^{1}\times\{\frac{1}{2}\}italic_α = italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT × { divide start_ARG 1 end_ARG start_ARG 2 end_ARG }. Let βݛ½\betaitalic_β be a circle between αݛ¼\alphaitalic_α and γݛ¾\gammaitalic_γ which is oriented clockwise.

Assumption 2.3.

Choose points x1,⋯,xnsubscriptݑ¥1normal-⋯subscriptݑ¥ݑ›x_{1},\cdots,x_{n}italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT sitting between γݛ¾\gammaitalic_γ and αݛ¼\alphaitalic_α. By braid isotopy, which preserves the transverse knot class (Theorem 2.8-(2)), we may assume that:

b∩A0={x1,⋯,xn}.ݑsubscriptݐ´0subscriptݑ¥1⋯subscriptݑ¥ݑ›b\cap A_{0}=\{x_{1},\cdots,x_{n}\}.italic_b ∩ italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = { italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } .

Let σisubscriptݜŽݑ–\sigma_{i}italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT (i=1,⋯,n-1ݑ–1⋯ݑ›1i=1,\cdots,n-1italic_i = 1 , ⋯ , italic_n - 1) be the generators of Artin’s braid group Bnsubscriptݐµݑ›B_{n}italic_B start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT satisfying σiσi+1σi=σi+1σiσi+1subscriptݜŽݑ–subscriptݜŽݑ–1subscriptݜŽݑ–subscriptݜŽݑ–1subscriptݜŽݑ–subscriptݜŽݑ–1\sigma_{i}\ \sigma_{i+1}\ \sigma_{i}=\sigma_{i+1}\ \sigma_{i}\ \sigma_{i+1}italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT and σiσj=σjσisubscriptݜŽݑ–subscriptݜŽݑ—subscriptݜŽݑ—subscriptݜŽݑ–\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT for |i-j|≥2ݑ–ݑ—2|i-j|\geq 2| italic_i - italic_j | ≥ 2. Geometrically, σisubscriptݜŽݑ–\sigma_{i}italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT acts by switching the marked points xisubscriptݑ¥ݑ–x_{i}italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and xi+1subscriptݑ¥ݑ–1x_{i+1}italic_x start_POSTSUBSC

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