Algebraic surface knots

In reality a physical knot is not a 1-dimensional object, but a solid object having a knotted torus as surface. On the other hand, we can study affine algebraic surfaces in R^3 which are given by one polynomial equation p(x,y,z)=0. In general such a surface can have singularities and can be non-compact. We study the construction of a knot as an algebraic knotted torus which is always possible. For example, the trefoil knot is given by a polynomial p of degree 14, which can be visualized by the SURFER software of Oberwolfach. One can ask what is the minimal degree of p to represent a given knot in this way, which defines a new knot invariant. We show that it can be related to the PL-degree of a knot, i.e. the minimal number of solid sticks which is needed to create a piecewise linear version of the knot.