Trinomials, torus knots and chains

Abstract

Letn>mn>mbe fixed positive coprime integers. Forv>0v>0, we give a topological description of the setΛ(v)\Lambda (v), consisting of points[x:y:z][x:y:z]in the complex projective plane for which the equationxζn+yζm+z=0x\zeta ^n +y \zeta ^m+z=0has a root with normvv. It is shown that the setΩ(v)=PC2∖Λ(v)\Omega (v)= {\mathbb P_{\mathbb C}} ^2 \setminus \Lambda (v)hasn+1n+1components. Moreover, the topological type of each component is given. The same results hold forΛ\LambdaandΩ=PC2∖Λ\Omega ={\mathbb P_{\mathbb C}}^2 \setminus \Lambda, whereΛ\Lambdadenotes the set obtained as the union of all the complex tangent lines to the33-sphere at the points of the torus knot, that is, the knot obtained by intersecting{[x:y:1]∈PC2:|x|2+|y|2=1}\{[x:y:1] \in \mathbb {P}_{\mathbb C}^2 : |x|^2+|y|^2=1\}and the complex curve{[x:y:1]∈PC2:ym=xn}\{[x:y:1] \in {\mathbb P_{\mathbb 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 ofΩ\Omegain a unique way.