Trinomials, torus knots and chains

• Published in: Transactions of the American Mathematical Society 2023-04, Vol.376 (4), p.2963
• Main Authors: Waldemar Barrera, Julio C. Magaña, Juan Pablo Navarrete
• Format: Article
• Language: English
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/tran/8834

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.