Trinomials, torus knots and chains

Abstract:

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 $\Omega (v)= {\mathbb P_{\mathbb C}} ^2 \setminus \Lambda (v)$ has $n+1$ components. Moreover, the topological type of each component is given. The same results hold for $\Lambda$ and $\Omega ={\mathbb P_{\mathbb C}}^2 \setminus \Lambda$, 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 $\{[x:y:1] \in \mathbb {P}_{\mathbb C}^2 : |x|^2+|y|^2=1\}$ and the complex curve $\{[x:y:1] \in {\mathbb P_{\mathbb C}} ^2 : y^m=x^n\}$. 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 of $\Omega$ in a unique way.