Invariants of embeddings of 2-surfaces in 3-space

TL;DR

This paper investigates the Seifert bilinear form as an isotopy invariant for embeddings of 2-surfaces in 3-space, providing characterizations of realizable forms, especially for the torus, and offers an accessible exposition.

Contribution

It characterizes the Seifert form invariants of surface embeddings and provides a simplified, accessible explanation of these results, including realizability conditions.

Findings

The Seifert form is $ ext{cap}$-symmetric for embeddings.

Any $ ext{cap}$-symmetric form is realizable for surfaces with boundary.

Characterization of realizable forms for the torus.

Abstract

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., $L(f)(\beta,\gamma)-L(f)(\gamma,\beta)=\beta\cap\gamma$ for any $\beta,\gamma\in H_1$. 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.