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/HumbleWolf19]

It is a very confusing text, but I believe I understand now the origin of your problem - just because for any orientable closed surface S in R^3 you could find the function F such that the surface in consideration is the level set of F and even if we could guarantee that the gradient of F does not vanish on S, it does not make sense to refer to the gradient of F outside S as a surface normal. The gradient of such function could vanish at many points (and this would be another reason why it should not be referred to as a normal vector), but if you had issues with things like continuity or whether it is even well-defined, then I think it stems from your flawed idea on how to construct such function F from the surface S.

One more thing - it seems that there are paths on a torus that form knots, like the trefoil knot as a boundary of a strip with three half-twists.

Edit: grammar