2014/09/24 by Tange, Motoo · 1 citation
#57M25 #57M27 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1409.7032
Ozsváth-Szabó proved the property that any coefficient of Alexander polynomial of lens space knot is either ±1 or 0 and the non-zero coefficients are alternating. Combining the formulas of the Alexander polynomial of lens space knots due to Kadokami-Yamada and Ichihara-Saito-Teragaito, we refine Ozsváth-Szabó's property as the existence of simple curves included in a region in \Bbb R2. The existence of curves, that has no end-points connected, is just 1-component in a region, can search distribution of non-zero coefficients of the Alexander polynomial of the lens space knot. This curve is much useful to obtain constraints of Alexander polynomials of lens space knots. For example, we can investigate the location of the second, third and fourth non-zero coefficients. The curve extracts new invariant α-index. The invariant is an important factor to determine Alexander polynomial of lens space knot. We classify lens space surgeries that the Alexander polynomial is the same as a (2,r)-torus knot and lens space surgeries with small genus and so on. As well as lens space knots in S3, we also deal with lens space knots in homology spheres, which the surgery duals are simple (1,1)-knots.