  In this paper, we derive formulae for the determinant of weaving knots
$W(3,n)$ and $W(p,2)$. We calculate the dimension of the first homology group
with coefficients in $\mathbb{Z}_3$ of the double cyclic cover of the
$3$-sphere $S^3$ branched over $W(3,n)$ and $W(p,2)$ respectively. As a
consequence, we obtain a lower bound of the unknotting number of $W(3,n)$ for
certain values of $n$.
