论文标题
在$ 3 $ -sphere中的表面不变的表面上
On invariants of surfaces in the $3$-sphere
论文作者
论文摘要
在本文中，我们研究了标准$ 3 $ -sphere中的封闭连接的定向表面的同位素类别。这样的表面将$ 3 $ -SPHERE拆分为两个紧凑的连接子手势，通过使用其Heegaard分裂，我们获得了$ 2 $ - 组成的句柄链接。在本文中，我们首先证明了这种2组分句柄链接的等效类，直到附加琐碎的$ 1 $ - handles可以恢复原始表面。因此，我们可以将$ 3 $ -sphere的表面的研究减少到$ 2 $ - 组成的句柄链接到稳定。然后，通过使用$ g $ - fomilies of Quandles，我们构建了$ 2 $ - 组成的句柄链接的不变性，以附加琐碎的$ 1 $ - handles，这导致$ 3 $ -sphere的表面不变。为了查看不变性的有效性，我们还将表明，我们的不变式可以区分$ 3 $ -sphere中的某些明确表面。
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.
