Unknotted homology classes on unknotted surfaces in $S^ 3$

Abstract:

Suppose $F$ is a closed, genus $g$ surface which is standardly embedded in ${S^3}$. Let $\gamma$ denote a primitive element in ${H_1}(F)$ which satisfies ${\theta _F}(\gamma ,\gamma ) = 0$ where ${\theta _F}$ is the Seifert pairing on $F$. We obtain a number theoretic condition which is equivalent to $\gamma$ being realizable by a curve (in $F$) which is unknotted in ${S^3}$. Various related observations are included.