  In this paper we compute the Kauffman bracket skein module of the complement
of $(2, 2p+1)$-torus knots, $KBSM(T_{(2, 2p+1)}^c)$, via braids. We start by
considering geometric mixed braids in $S^3$, the closure of which are mixed
links in $S^3$ that represent links in the complement of $(2, 2p+1)$-torus
knots, $T_{(2, 2p+1)}^c$. Using the technique of parting and combing, we obtain
algebraic mixed braids, that is, mixed braids that belong to the mixed braid
group $B_{2, n}$ and that are followed by their ``coset'' part, that represents
$T_{(2, 2p+1)}^c$. In that way we show that links in $T_{(2, 2p+1)}^c$ may be
pushed to the genus 2 handlebody, $H_2$, and we establish a relation between
$KBSM(T_{(2, 2p+1)}^c)$ and $KBSM(H_2)$. In particular, we show that in order
to compute $KBSM(T_{(2, 2p+1)}^c)$ it suffices to consider a basis of
$KBSM(H_2)$ and study the effect of combing on elements in this basis. We
consider the standard basis of $KBSM(H_2)$ and we show how to treat its
elements in $KBSM(T_{(2, 2p+1)}^c)$, passing through many different spanning
sets for $KBSM(T_{(2, 2p+1)}^c)$. These spanning sets form the intermediate
steps in order to reach at the set $\mathcal{B}_{T_{(2, 2p+1)}^c}$, which,
using an ordering relation and the notion of total winding, we prove that it
forms a basis for $KBSM(T_{(2, 2p+1)}^c)$. We finally consider c.c.o.
3-manifolds $M$ obtained from $S^3$ by surgery along the trefoil knot and we
discuss steps needed in order to compute the Kauffman bracket skein module of
$M$. We first demonstrate the process described before for computing the
Kauffman bracket skein module of the complement of the trefoil, $KBSM(Tr^c)$,
and we study the effect of braid band moves on elements in the basis of
$KBSM(Tr^c)$. These moves reflect isotopy in $M$ and are similar to the second
Kirby moves.
