On the reflexivity of certain fibered 3-knots (Q1202226)

For a matrix \(A\) in \(A\) in \(\text{SL}(n,\mathbb{Z})\), \(n>2\), Cappell and Shaneson showed that, under a certain condition (CS), the mapping torus of a diffeomorphism on a torus \(T^ n\) induced by \(A\) is a homology \(S^ 1\times S^ n\) which, up to isotopy, has a unique loop whose conjugates generate the fundamental group. By surgery on the loop one gets a pair of \((n-1)\)-knots in homotopy \((n+1)\)-spheres according to the 2 possible framings of the loop. If these knots are equivalent then the knot is called reflexive and is determined by its exterior. Cappell and Shaneson gave a certain condition on the matrix \(A\) characterizing reflexivity of the associated knot (thereby constructing inequivalent knots with the same complement). In the present note, generalizing results of Hillman-Wilson, the case \(n=4\) is studied in detail: the occuring matrices \(A\) satisfying (CS) are characterized by their characteristic polynomials, and reflexivity of the associated knots is decided using this characterization.