  In this paper we study embeddings of oriented connected closed surfaces in
$\mathbb S^3$. We define a complete invariant, the fundamental span, for such
embeddings, generalizing the notion of the peripheral system of a knot group.
From the fundamental span, several computable invariants are derived and
employed to study handlebody knots, bi-knotted surfaces, and chirality of
knots. These invariants are capable to distinguish inequivalent handlebody
knots and bi-knotted surfaces with homeomorphic complements. Particularly, we
obtain an alternative proof of the inequivalence of Ishii et al.'s handlebody
knots $5_{1}$ and $6_{4}$, and also construct an infinite family of pairs of
inequivalent bi-knotted surfaces with homeomorphic complements. An
interpretation of Fox's invariant in terms of the fundamental span is discussed
and used to show $9_{42}$ and $10_{71}$ in the Rolfsen knot table are chiral;
their chirality is known to be undetectable by the Jones and HOMFLY-PT
polynomials.
