TRINOMIALS, TORUS KNOTS AND CHAINS

Let n > m be fixed positive coprime integers. For v > 0, we give a topological description of the set lambda(v), consisting of points [x : y : z] in the complex projective plane for which the equation x zeta(n) + y zeta(m) + z = 0 has a root with norm v. It is shown that the set 2(v) = P-C(2) \ lambda(v) has n+ 1 components. Moreover, the topological type of each component is given. The same results hold for lambda and omega = P-C(2) \ lambda, where lambda denotes the set obtained as the union of all the complex tangent lines to the 3-sphere at the points of the torus knot, that is, the knot obtained by intersecting {[x : y : 1] is an element of P-C(2) : |x|2 + |y|2 = 1} and the complex curve {[x : y : 1] is an element of P-C(2) : y(m) = x(n)}. Finally, we use the linking number of a distinguished family of circles and the torus knot to give a numerical invariant which determines the components of 2 in a unique way.