Open book foliation

Abstract

We study open book foliations on surfaces in 3-manifolds and give applications to contact geometry of dimension 3. We prove a braid-theoretic formula for the self-linking number of transverse links, which reveals an unexpected connection with to the Johnson-Morita homomorphism in mapping class group theory. We also give an alternative combinatorial proof of the Bennequin-Eliashberg inequality.

On the self-linking number of transverse links

“…Since S has connected boundary the separating property implies that x · [C i ] = 0 for any x ∈ H 1 (S, ∂S). Moreover, by [8,Proposition 3.12- (4)] we have c([T C i ], x) = 0. Repeatedly using the crossed homomorphism property of the function c, proven in [8, Proposition 3.12- (2)], we have • each of the n points is joined with C by an a-arc in the foliation F ob (Σ).…”

“…In particular, the proof of [8,Claim 3.8] implies that b ψ is also null-homologous. Recall the Johnson kernel K g , the kernel of the Johnson homomophism τ :…”

“…Let Σ and Σ ψ be Seifert surfaces constructed by using the above a ∈ H 1 (S, ∂S) and the method as in the proof of [8,Theorem 3.10]. If ψ ∈ K g , then…”

“…, ρ 2g+r−1 } of the braid group as depicted in Figure 1. It is implicit in the proof of [8,Claim 3.8] that the homology class a ∈ H 1 (S, ∂S; Z) is represented by properly embedded circles and arcs that do not intersect the n punctures of S. Since φ = id near ∂S we may regard a − φ * (a) ∈ H 1 (S; Z).…”

“…See [8,Definition 3.3]) for the precise definition of the normal form, where we use the notation N (a) instead of a NF . Let F be a cobordism surface from φ(a NF ) to (φ * (a)) NF , i.e., a compact, oriented, properly embedded surface in S × [0, 1] such that…”