On cyclic branched coverings of torus knots

We prove that the n-fold cyclic coverings of the 3-sphere branched over the torus knots K(p, q), p > 1 ≥ 2 (i.e. the Brieskorn manifolds in the sense of [12]) admit spines corresponding to cyclic presentations of groups if p = 1 (mod q). These presentations include as a very particular case the Sieradski groups, first introduced in [14] and successively obtained from geometric constructions in [4], [9], and [15]. So our main theorem answers in affirmative to an open question suggested by the referee in [14]. Then we discuss a question concerning cyclic presentations of groups and Alexander polynomials of knots. © Birkhäuser Verlag, Basel, 1999.