  Ozsv\'ath-Szab\'o proved the property that any coefficient of Alexander
polynomial of lens space knot is either $\pm1$ 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 Ozsv\'ath-Szab\'o's property as the
existence of simple curves included in a region in ${\Bbb R}^2$. 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 $\alpha$-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
$S^3$, we also deal with lens space knots in homology spheres, which the
surgery duals are simple (1,1)-knots.
