Uniqueness of Seifert surfaces of knots

Question

I know the theorem that Given a knot K in the 3-sphere, it has a Seifert surface S whose boundary is K. So, can we also say that for every unique Seifert surface there is an unique knot and vice versa?

Answer 1

Every Seifert surface $S$ has a unique associated knot $K$ (because $K$ is by definition just the boundary of $S$) but the converse is not true. There are certainly knots with more than one associated Seifert surface as a Seifert surface of a knot is dependant on the knot diagram being used for $K$ (when applying the standard Seifert algorithm anyway). This is why the genus of a knot is defined to be the minimum genus of all Seifert surfaces of the knot. Given a knot with a Seifert surface of genus $g$, it is always possible to construct a seifert surface of the same knot with genus $g+1$ using surgery as described here.

Answer 2

There is an operation on Seifert surfaces defined by taking an arbitrary arc from the surface to itself with the arc interior embedded in the complement of the surface, and then adding a small tube to the surface along the arc. You can make an arbitrarily complicated Seifert surface for any knot this way. Perhaps more surprisingly, any two Seifert surfaces for the same knot are related to each other by a sequence of such moves and their inverses. So the Seifert surface is unique up to this equivalence relation.