Narrow Results

In 1998, C. Lescop proved that any integral homology sphere with the Casson invariant zero can be obtained from S³ by surgery on a boundary link each component of which has a trivial Alexander polynomial. In this paper, we prove that for any integral homology sphere H, there exists an integer k such that H can be obtained from S³ by surgery on a boundary link each component of which has the Alexander polynomial 1+k(t^½-t^-½)².

By obtaining surgery descriptions of knots which lie on the genus one fiber of the trefoil or figure eight knot, we show that these include hyperbolic knots with arbitrarily large volume. These knots admit lens space surgeries and form two families of Berge knots. By way of tangle descriptions we also obtain surgery descriptions for these knots on minimally twisted chain links.

Using Kirby calculus, we explicitly pass from Berge's R-R descriptions of ten families of knots with lens space surgeries to surgery descriptions on the minimally twisted five chain link (MT5C). Since the MT5C admits a strong inversion, we also give the corresponding tangle descriptions.