I Do there exist surfaces whose boundary is a closed knot?

[jk22]

I ask this for the condition of application of Stoke's theorem.

 

[Infrared]

Every knot is the boundary of an orientable (you want this for Stokes' theorem) surface. Look up "Seifert surface".

 

[lavinia]

https://homepages.warwick.ac.uk/~masgar/Talks/Simons/minimal_and_seifert_surfaces.pdf

 

[zinq]

Those slides mention that Lawson, in his dissertation (Annals of Mathematics, November 1970), proved that "all genera" can be realized as minimal surfaces in the 3-sphere.

In fact, Lawson shows that all compact orientable surfaces exist as embedded minimal surfaces in S³, and that all compact non-orientable surfaces but one can occur as immersed minimal surfaces in S³. The exception is the projective plane, which Lawson proves cannot occur as a minimal immersed surface in S³.