On closed oriented surfaces in the 3-sphere

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.