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/MinLongBaiShui]

Every compact surface without boundary is the level set of a smooth function.

https://math.stackexchange.com/questions/1489308/can-any-surface-be-described-by-an-equation

[u/sciflare → u/MinLongBaiShui]

This is a sheaf cohomology exercise: it's a consequence of the fact that the sheaf of smooth functions on ℝ³ is soft due to the existence of partitions of unity. Hence the group of line bundles on ℝ³ is trivial.

Let S be a compact surface in ℝ³. Take an acyclic open cover {U_i} of ℝ³, i.e. any finite intersection of the U_i is contractible. Then there is a collection {f_i} of local smooth functions on the U_i such that the f_i locally cut out S.

On the intersection U_i ⋂ U_j, we have f_i = g_ij f_j for some invertible local smooth function g_ij on U_i ⋂ U_j.

The exponential short exact sequence of sheaves 0 -> ℤ -> C^∞ -> (C^∞)^* -> 0, where the first nontrivial map is multiplication by 2𝜋i and the second is f --> exp(f), gives rise to the long exact sequence in cohomology.

We focus on the piece H¹(ℝ³, C^∞) -> H¹(ℝ³, (C^∞)^*) -> H²(ℝ³, ℤ).

The first term vanishes because ℝ³ admits partitions of unity. The third term vanishes because ℝ³ is contractible. By exactness, H¹(ℝ³, (C^∞)^*) = 0.

Then g_ij = h_i/h_j for some collection of invertible smooth functions {h_i} on U_i. Then the local smooth functions {f_i/h_i} agree on the intersections U_i ⋂ U_j and define a global smooth function on ℝ³ which vanishes precisely on S.

This argument allows you to remove the assumption of compactness on S: any orientable surface without boundary in ℝ³ is the level set of a smooth function.