On maximal fibred submanifolds of a knot exterior (Q1106511)

Let K be a knot in the 3-sphere \(S^ 3\), E its exterior and \(G=\pi_ 1(E)\) the group of K. For an incompressible spanning surface \(S\subset E\) for K, we define a subgroup \(\mu\) (S) of G by \(\mu (S)=\cap_{k\in {\mathbb{Z}}}m^ k\pi_ 1(S)m^{-k}\) where m is the element of G corresponding to a meridian. We prove that \(\mu\) (S) is an invariant for K and it is realized by a compact surface m(S) on S; in particular \(\mu\) (S) is a finitely generated free group. Furthermore it is shown that there is a fibred submanifold F(S) of W with fibre m(S) which is maximal in a sense, and that the pair (F(S),m(S)) is unique up to ambient isotopy of E.