# Curves defined by Chebyshev polynomials - Mathematics > Algebraic Geometry

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Abstract: Working over a field $\kk$ of characteristic zero, this paper studies lineembeddings of the form $\phi = T i,T j,T k:\A^1\to\A^3$, where $T n$ denotesthe degree $n$ Chebyshev polynomial of the first kind. In {\it Section 4}, itis shown that 1 $\phi$ is an embedding if and only if the pairwise greatestcommon divisor of $i,j,k$ is 1, and 2 for a fixed pair $i,j$ of relativelyprime positive integers, the embeddings of the form $T i,T j,T k$ represent afinite number of algebraic equivalence classes. {\it Section 2} gives analgebraic definition of the Chebyshev polynomials, where their basic identitiesare established, and {\it Section 3} studies the plane curves $T i,T j$. {\itSection 5} establishes the Parity Property for Nodal Curves, and uses this toparametrize the family of alternating $i,j$-knots over the real numbers.

Author: ** Gene Freudenburg, Jenna Freudenburg**

Source: https://arxiv.org/