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Ozsv\'ath-Szab\'o proved the property that any coefficient of Alexanderpolynomial of lens space knot is either $\pm1$ or $0$ and the non-zerocoefficients are alternating. Combining the formulas of the Alexanderpolynomial of lens space knots due to Kadokami-Yamada andIchihara-Saito-Teragaito, we refine Ozsv\'ath-Szab\'o's property as theexistence of simple curves included in a region in ${\Bbb R}^2$. The existenceof 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 ofthe lens space knot. This curve is much useful to obtain constraints ofAlexander polynomials of lens space knots. For example, we can investigate thelocation of the second, third and fourth non-zero coefficients. The curveextracts new invariant $\alpha$-index. The invariant is an important factor todetermine Alexander polynomial of lens space knot. We classify lens spacesurgeries that the Alexander polynomial is the same as a $(2,r)$-torus knot andlens space surgeries with small genus and so on. As well as lens space knots in$S^3$, we also deal with lens space knots in homology spheres, which thesurgery duals are simple (1,1)-knots.

On the Alexander polynomial of lens space knot