On invariants of surfaces in the 3-sphere

In this paper we study isotopy classes of closed connected orientable surfaces in the standard 3-sphere. Such a surface splits the 3-sphere into two compact connected submanifolds, and by using their Heegaard splittings, we obtain a 2-component handlebody-link. In this paper, we first show that the equivalence class of such a 2-component handlebody-link up to attaching trivial 1-handles can recover the original surface. Therefore, we can reduce the study of surfaces in the 3-sphere to that of 2-component handlebody-links up to stabilizations. Then, by using G-families of quandles, we construct invariants of 2-component handlebody-links up to attaching trivial 1-handles, which lead to invariants of surfaces in the 3-sphere. In order to see the effectiveness of our invariants, we will also show that our invariants can distinguish certain explicit surfaces in the 3-sphere.