let $K$ be an $S^1$ knot, i.e., $K$ is a homeomorphic copy of the "standard" $S^1$ living in $\mathbb R^3$ ). I am trying to see if $K$ is orientable. This is what I have: 1) If the map sending $S^1$ to $K$ is a diffeomorphism, we are done. Diffeomorphisms are either orientation preserving or orientation reversing. Besides , a diffeomorphism would give rise to a vector space (tangent space) isomorphism, and then the image tangent space describes an orientation.
2) If the embedding is not a diffeomorphism, just a homeomorphism. Then maybe we could see what happens with the image of the top (co) homology with coefficients in $\mathbb Z$. The induced maps on (co)homology are isomorphisms. So the top class, the orientation/fundamental class cannot be mapped to the $0$ class. But it will be mapped (I think) to a subgroup of $\mathbb Z$ , meaning to {$ n \mathbb Z$}. Am I on the right track? Thanks.