论文标题
2曲面的嵌入在3空间中的嵌入
Invariants of embeddings of 2-surfaces in 3-space
论文作者
论文摘要
令$ m $为带有手柄和孔的球体，$ f：m \ to \ mathbb r^3 $ a嵌入，$ h_1 = h_1（m; \ \ mathbb z）$。我们研究了$ f $的简单同位素不变，Seifert双线性表格$ l（f）：h_1 \ times h_1 \ to \ mathbb z $。令$ \ cap：h_1 \ times h_1 \ to \ mathbb z $是$ m $的交点。然后，Seifert形式为$ \ cap $ -smmetric，即$ l（f）（β，γ）-L（f）（γ，β）=β\capγ$，用于任何$β，γ\ in h_1 $。如果$ m $具有非空边界，则任何$ \ cap $ -smmetric双线性表格$ h_1 \ times h_1 \ to \ mathbb z $可将某些嵌入$ f $的$ l（f）$实现。我们为圆环$ m $介绍了可实现的表格的特征。结果很简单，并且可能在民间传说中闻名。我们提出了一个简化的非专家访问的简化博览会。
Let $M$ be a sphere with handles and holes, $f:M\to\mathbb R^3$ an embedding, and $H_1=H_1(M;\mathbb Z)$. We study a simple isotopy invariant of $f$, the Seifert bilinear form $L(f):H_1\times H_1\to\mathbb Z$. Let $\cap:H_1\times H_1\to\mathbb Z$ be the intersection form of $M$. Then the Seifert form is $\cap$-symmetric, i.e., $L(f)(β,γ)-L(f)(γ,β)=β\capγ$ for any $β,γ\in H_1$. If $M$ has non-empty boundary, then any $\cap$-symmetric bilinear form $H_1\times H_1\to\mathbb Z$ is realizable as $L(f)$ for some embedding $f$. We present a characterization of realizable forms for the torus $M$. The results are simple and presumably known in folklore. We present a simplified exposition accessible to non-specialists.
