On the Alexander polynomial of lens space knots

Abstract

Ozsváth and Szabó proved the property that coefficients of the Alexander polynomial of any lens space knot are either ±1 or 0 and the non-zero coefficients are alternating. Combining the formulas of the Alexander polynomial of lens space knots by Kadokami and Yamada, and Ichihara, Saito and Teragaito, we refine Ozsváth and Szabó's property as the existence of simple curves included in a region in R2. The curves have no end-points and include just 1-component in a region. This can be much useful for studying the distribution of non-zero coefficients of the Alexander polynomial of the lens space knot. For example, we can investigate the location of the second, third and fourth non-zero coefficients. We extract a new invariant α-index. This invariant is an important factor to determine Alexander polynomials of lens space knots. We give a criterion for a lens space surgery that the Alexander polynomial is the same as a (2,r)-torus knot and a lens space surgery with small genus and so on. As well as lens space knots in S3, we also deal with lens space knots in homology spheres whose surgery duals are simple (1,1)-knots.