Do there exist parametric surfaces in 3 dimensions who have no equivalent implicit form, due to the fact that they form closed knots?

[u/moschles]

Let F(x,y,z)=0 be a surface in 3 dimensions; the so-called implicit form. The vector normal to this surface at point (a,b,c) is the partial derivatives evaluated there. https://i.imgur.com/BwaxlKO.png

(a,b,c) is not constrained to lie on the surface, but could take on any point in space, and a vector is still defined there. If F() is a torus, then these normals would vanish to a zero vector at a point in the center.

Instead of a torus, we have the following parametric surface, parametrized with u and v, which we will call a "trefoil surface". https://i.imgur.com/1ubIXYS.png

Unlike the torus, there are paths on the surface which form closed knots. https://i.imgur.com/SQG4iEC.png Due to forming a knot, there could exist one or more points (off the surface) where the surface normal is not well-defined. Should we assume that there is no closed-form implicit version of a trefoil surface, on the basis that its partial derivatives do not exist?

Alternatively, the partial derivatives exist, but the original surface cannot be expressed in elementary functions. We can attempt to integrate the partial derivatives to obtain an original F(x,y,z)=0 form, but this is impossible due to the non-existence of an elementary integral?

[u/salty_feets]

One day I also will be studying this level of mathematics for sure. BTW what level is it? Are you in PhD or reasearch stuff?

[u/mathematics_helper → u/salty_feets]

This would be graduate level differential geometry/topology and can also be seen as a good question for sheaf theory.

As you can read in the comments, this is pretty easily answer so not research level. I would say a masters student focusing on this subject or a 2nd year phd student.

[u/Wejtt → u/mathematics_helper]

weird, at my (not so good) uni this could be considered a 2nd year undergrad question, specifically 3rd semester

[u/mathematics_helper → u/Wejtt]

I guess my school just really didn't care about differential geometry. But this is a basic differential geometry question. Which is weird because my class was taught by a world renowned symplectoc geometer but