Curves defined by Chebyshev polynomials

2009/02/19 by Freudenburg, Gene, Freudenburg, Jenna
#14H50 #14R10 #57M25 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.0902.3440

Abstract

Working over a field \kk of characteristic zero, this paper studies line embeddings of the form ϕ= (T_i,T_j,T_k):\A¹→\A³, where T_n denotes the degree n Chebyshev polynomial of the first kind. In \it Section 4, it is shown that (1) ϕ 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.

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