Curves defined by Chebyshev polynomials

Abstract

Working over a field $\kk$ of characteristic zero, this paper studies line embeddings of the form $\phi = (T_i,T_j,T_k):\A^1\to\A^3$, where $T_n$ denotes the degree $n$ Chebyshev polynomial of the first kind. In {\it Section 4}, it is shown that (1) $\phi$ is an embedding if and only if the pairwise greatest common divisor of $i,j,k$ is 1, and (2) for a fixed pair $i,j$ of relatively prime positive integers, the embeddings of the form $(T_i,T_j,T_k)$ represent a finite number of algebraic equivalence classes. {\it Section 2} gives an algebraic definition of the Chebyshev polynomials, where their basic identities are established, and {\it Section 3} studies the plane curves $(T_i,T_j)$. {\it Section 5} establishes the Parity Property for Nodal Curves, and uses this to parametrize the family of alternating $(i,j)$-knots over the real numbers.