It had been known that the normal closure of the pure braid group $P_n(D)$ of the disk in the pure braid group of the torus $P_n(T)$ is the commutator subgroup. I am going to study the case of full braid groups: i.e., the normal closure of $B_n(D)$ in $B_n(T)$, which turns out to have an interesting geometric description.