In knot theory, we know that same linking number cannot distinguish two different knots/links. For example, whitehead link(linking number$=0$) and unlink of 2 components (linking number$=0$) but whitehead link is not tricolourable while unlink of 2 componetns is tricolourable. Is genus of a link an invariant? For example, if I have a genus of a link is different from the genus of an unlink, can I say that they are not equivalent?
Yes, genus is a link invariant. This means that isotopic links have the same genus, or the contrapositive, that links with different genera cannot be isotopic.
Unfortunately, calculating the genus can be difficult, as one has to minimize the genus over all Seifert surfaces.