Research ArticleNo Access

On invariants of surfaces in the 3-sphere

Hiroaki Kurihara

Artificial Intelligence Laboratory, Fujitsu Research, Fujitsu, Japan

E-mail Address: kurihara.hiroaki.766@gmail.com

https://doi.org/10.1142/S0218216523500463Cited by:0 (Source: Crossref)

Abstract

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.

Keywords:

AMSC: 57M99

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