Math. Ann. 287, 259--273 (1990) Ilallmmlitcll0 Am 9 Springer-Verlag 1990 Geometrically fibred two-knots J. A. Hillman, t'* and S. P. Plotnick 2 i School of Mathematics, Physics, Computing and Electronics, Macquire University, Macquarie, N.S.W. 2109, Australia 2 Department of Mathematics, State Universityof New York at Albany, Albany,NY 12222,USA The most familiar invariants of knots are derived from the knot complements, and so it is natural to ask whether every knot is determined by its complement. Gluck showed that given a 2-knot there is at most one other 2-knot with the same complement I-G1]. (This was subsequently shown to hold also in all dimensions). A knot which is determined by its complement is said to be reflexive. The first examples of 2-knots which are not reflexive were given by Cappell and Shaneson, these were fibred knots with closed fibre •3/7/3 [CS]. (Their argument should work in all higher dimensions, but examples of nonreflexive n-knots are only known for n = 2, 3 or 4.) Gordon gave a different family of examples [Go], and Plotnick extended his result to show that no fibred 2-knot with monodromy of odd order is reflexive [P3]. It is plausible that this may be so whenever the order is greater than 2, but this is at present unknown. We shall consider 2-knots which are fibred with closed fibre an irreducible 3-manifold with a geometric structure in the sense of Thurston [S, T1]. A nontrivial cyclic branched cover of S 3, branched over a knot, admits a geometric structure if and only if the knot is a prime, simple knot. The geometry is then Sl'., S 3, H 3, E 3 or Nil. In Theorem 1 we shall show that no cyclic branched cover of the r-twist spin of such a knot is ever reflexive, if r > 2. (Our argument also explains why fibred knots with monodromy of order 2 are reflexive.) If the 3-dimensional Poincar6 conjecture is true then all fibred 2-knots with monodromy of finite order are cyclic branched covers of twist spins of classical knots. If the closed fibre has geometry E 3 then it must be either R3/Z 3 or the fiat orientable 3-manifold with noncyclic holonomy, with fundamental group G6. (Cf. [W, p. 117] and Theorem 4 of [H2]. Note that the subsequent remark that the centre of the knot group G( + ) is trivial is wrong.) The only such knots with monodromy finite order greater than 2 are the 3-twist spin of the figure eight knot and its 2-fold cyclic branched cover, and these are not reflexive, by [Go]. (They are Gluck reconstructions of each other.) * Current address: Department of Pure Mathematics, University of Sydney,Sydney, N.S.W. 2006, Australia