Say p, q are positive integers with gcd(p,q) = 1 and p > q. Then a (p,q) torus knot in S^3 in R^4 = C^2 = {(z,w)} is given by the intersection of the 3-sphere S^3 |z|^2 + |w|^2 = 1 and the variety z^p + w^q = 0. Via stereographic projection S^3 - {(0,0,0,1)} —> R^3 = {(X,Y,Z)}, this gives the complex polynomial equation P(X,Y,Z) = 0 where P(X,Y,Z) = 2^p (X+iY)^p + (Q+1)^(p-q) (2Z + i(Q-1))^q and Q stands for X^2 + Y^2 + Z^2, which is equivalent to the two real polynomial equations Re(P(X,Y,Z)) = 0 and Im(P(X,Y,Z) = 0, each of degree = 2p. This is the lowest degree for two real polynomials that give the (p,q) torus knot in R^3 (p > q). Of course we could write (Re("))^2 + (Im("))^2 = 0 and get the same knot as the zero-set of just one real polynomial of degree 4p. —Dan