Computing Chebyshev knot diagrams. (May 2018)

• Record Type: Journal Article
• Authors: Koseleff, P.-V.; Pecker, D.; Rouillier, F.; Tran, C.
• Is Part Of: Journal of symbolic computation. Volume 86(2018)
• Page Start: 120
• Page End: 141
• Publication Date: 2018-05
• Subjects: 14H50 -- 68W30 -- 57M25; Zero dimensional systems -- Chebyshev curves -- Lissajous curves -- Lissajous knots -- Polynomial knots -- Chebyshev polynomials -- Minimal polynomial -- Chebyshev forms; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2017.04.001
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: A Chebyshev curve C ( a, b, c, φ ) has a parametrization of the form x ( t ) = T a ( t ) ; y ( t ) = T b ( t ) ; z ( t ) = T c ( t + φ ), where a, b, c are integers, T n ( t ) is the Chebyshev polynomial of degree n and φ ∈ R . When C ( a, b, c, φ ) is nonsingular, it defines a polynomial knot. We determine all possible knot diagrams when φ varies. When a, b, c are integers, ( a, b ) = 1, we show that one can list all possible knots C ( a, b, c, φ ) in O ˜ ( n 2 ) bit operations, with n = a b c . We give the parameterizations of minimal degree for all two-bridge knots with 10 crossings and fewer.