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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 groupwith 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 aconsequence, we obtain a lower bound of the unknotting number of $W(3,n)$ forcertain values of $n$.

Some evaluations of Jones polynomials for certain families of weaving knots