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

摘要

Let (∑、ø) denote an open book decomposition of a closed oriented 3-manifold where ∑ is the fiber surface with connected boundary and ø the monodromy map. In the present paper we investigate the Alexander polynomial of (∑、ø). We focus on algebraic intersection numbers of some essential arcs on ∑ with their images under ø. 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.