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


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].

DOI Code: 10.1285/i15900932v30n1p41

3–manifold; branched covering; orbifold; fundamental group; homology 3–sphere; (1, 1)-knot; torus knot

Classification: 57M05; 57M12; 57R65

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