A note on some homology spheres which are 2-fold coverings of inequivalent knots

Abstract

We construct a family of closed 3--manifolds $M_{\alpha,r}$, which are  homeomorphic to the Brieskorn homology spheres $\Sigma(2, \alpha+1, q+2r-1)$,  where $q=\alpha(r-1)$ and both $\alpha \ge 1$ and $q \ge 3$ are odd. We show  that $M_{\alpha,r}$ can be represented as 2--fold covering of the 3--sphere  branched over two inequivalent knots. Our proofs follow immediately from two  different symmetries of a genus 2 Heegaard diagram of $\Sigma(2, \alpha+1,  q+2r-1)$, and generalize analogous results proved in [BGM], [IK], [SIK] and  [T].