  For all natural numbers $N$ and prime numbers $p$, we find a knot $K$ whose
skein polynomial $P_K(a,z)$ evaluated at $z=N$ has trivial reduction modulo
$p$. An interesting consequence of our construction is that all polynomials
$P_K(a,N)$ (mod~$p$) with bounded $a$-span are realised by knots with bounded
braid index. As an application, we classify all polynomials of the form
$P_K(a,1)$ (mod $2$) with $a$-span $\leq 10$.
