Let $P$ be a polygon, and therefore a topological disk. Suppose we make some identifications on its edges, possibly identifying 2 or more edges of the polygon to a single edge, to get a 2-complex $K$ with a single 2-cell. Thus, by going around the boundary we get some word in the edge labels and their inverses. It is well known, for example, that $aba^{-1}b^{-1}$ would give the torus, $aa$ the projective plane, and $abac$ the mobius strip. Is there any way to tell easily, from this word, whether $K$ can be embedded in $S^3$?
I know it can theoretically be tested by classifying "thickenings" of $K$ and doing some pretty complicated stuff that works for any 2-complex, but I am interested in whether there is a simpler more combinatorial classification in this specific case where we have only a single 2-cell.
If it helps I do not mind assuming that there is only a single vertex, and that there are no subwords of the form $xx^{-1}$ or $x^{-1}x$ for any character $x$, since these assumptions do not seem harmful.