What is the relationship between the Euler number of Seifert surface of a link and its linking number

Question

Can I know whether the link is unlinked indeed (split Union of classical knots) from the Euler number of Seifert surface.

Answer 1

Seifert surfaces by definition are connected, so no: The Hopf link and 2-component unlink both have genus 0 Seifert surfaces (of Euler number 0, since they have two boundary components).

If you allow disconnected bounding surfaces (such that every component has boundary), then yes: an $n$-component link would have bounding surface of at most $n$ components, which then has Euler number at most $n$ (realized by $n$ discs). An n-component link is the unlink if and only if it bounds $n$ (disjoint, of course) discs, so it's the unlink if and only if it bounds a surface of Euler number $n$ with no boundaryless components.