Heegaard splittings of the Brieskorn homology spheres that are equivalent after one stabilization

Abstract

We construct a class of 3-manifolds $M_q$ which are homeomorphic to  the Brieskorn homology spheres $\sum(2,3,q)$, where $(2,3,q)$ are  relatively prime. Also, we show that $M_q$ is a $2$-fold cyclic  branched covering of $S^3$ over a knot $K_q$ which is inequivalent  with torus knot $T(3,q)$ for $q \ge 7$. Moreover, we show that two  inequivalent Heegaard splittings of $\sum(2,3,q)$ of genus 2  associated with $T(3,q)$ and $K_q$ are equivalent after single  stabilization.