On the Alexander polynomials of open book decompositions of 3-manifolds

Let (Σ,ϕ) denote an open book decomposition of a closed oriented 3-manifold where Σ is the fiber surface with connected boundary and ϕ the monodromy map. In the present paper we investigate the Alexander polynomial of (Σ,ϕ). We focus on algebraic intersection numbers of some essential arcs on Σ with their images under ϕ. We then give a description of the Alexander polynomial with a matrix of the intersection numbers. Moreover we will see that each coefficient of the Alexander–Conway polynomial of the open book decomposition can be expressed by Pfaffians of some matrices of the intersection numbers.