Computing Chebyshev knot diagrams. (May 2018) - Koseleff, P.-V.; Pecker, D.; Rouillier, F.; Tran, C.
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.