Trinomials, torus knots and chains

Resumen

Let n > m n>m be fixed positive coprime integers. For v > 0 v>0 , we give a topological description of the set Λ ( v ) \Lambda (v) , consisting of points [ x : y : z ] [x:y:z] in the complex projective plane for which the equation x ζ n + y ζ m + z = 0 x\zeta ^n +y \zeta ^m+z=0 has a root with norm v v . It is shown that the set Ω ( v ) = P C 2 ∖ Λ ( v ) \Omega (v)= {\mathbb P_{\mathbb C}} ^2 \setminus \Lambda (v) has n + 1 n+1 components. Moreover, the topological type of each component is given. The same results hold for Λ \Lambda and Ω = P C 2 ∖ Λ \Omega ={\mathbb P_{\mathbb C}}^2 \setminus \Lambda , where Λ \Lambda denotes the set obtained as the union of all the complex tangent lines to the 3 3 -sphere at the points of the torus knot, that is, the knot obtained by intersecting { [ x : y : 1 ] ∈ P C 2 : | 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 ] ∈ P C 2 : y m = x n } \{[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 Ω \Omega in a unique way.