Invariants of embeddings of 2-surfaces in 3-space

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 $\cap:H_1\times H_1\to\mathbb Z$betheintersectionformof$M$\mathrm{.}ThentheSeifertformis$\cap$-symmetric,i\mathrm{.}e\mathrm{.},$L(f)(\beta,\gamma)-L(f)(\gamma,\beta)=\beta\cap\gamma$forany$\beta,\gamma\in H_1$\mathrm{.}If$M$hasnon-emptyboundary,thenany$\cap$-symmetricbilinearform$H_1\times H_1\to\mathbb Z$isrealizableas$L(f)$forsomeembedding$f$. 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.