Classification of quartics possessing a non-simple singular point. II

The paper is devoted to comparing two classifications, up to rigid isotopy and up to $PL$-homeomorphism, of surfaces of degree 4 in $\mathbb{C}p^3$ (quartics) possessing at least one non-simple singular point. The main $PL$-invariant to distinguish quartics is the obvious lattice morphism $\oplus M(O_i)\oplus<4>\mapsto K3$, $M(O_i)$ being the Milnor lattices of all the singular points of the quartic and $K3=2E_8\oplus3U$ being the intersection lattice of a nonsingular quartic. The main result is the following theorem.
THEOREM. With the exception of several cases a quartic $V$ is determined up to rigid isotopy by the corresponding lattice morphism. The exceptions are some quartics with the singular set of the type $X_9+\sum A_{2p_i-1}+\sum D_{2q_j}$, $\sum p_i+\sum(q_j+1)$ being equal to 6 or 7.
Some auxiliary results of the paper also may be of interest: the relation between the Milnor lattice of a singularity and the lattice of its resolution is established. This provides algebraically clear description of the Milnor lattices of most singularities.