There exist inequivalent knots with the same complement

The study and classification of knots has been based upon invariants of the knot complement. In this paper we give examples of inequivalent smooth spherical knots with diffeomorphic complements. These examples will also not be reflections, inversions or reflected inversions of each other. The knots we consider can be described as the class of all fibred knots with fibre a punctured n-torus, n > 3, and with the monodromy map on first homology having no negative eigenvalues. With easy modifications the arguments show that the constructed examples are not even piecewise linearly or topologically equivalent. A smooth n-knot is a smooth submanifold K of the (n + 2)-sphere S"+2, homeomorphic to So, and if K is diffeomorphic to S", it is said that the