A polynomial parametrization of torus knots

2007/12/14 by Pierre-Vincent Koseleff, Daniel Pecker, Koseleff, Pierre-Vincent +1
Mathematics · #Advanced Combinatorial Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.HO

paper · pdf · doi:10.48550/arxiv.0712.2408

arxiv created 2007/12/14 · arxiv updated 2009/12/01

Abstract

For every odd integer N we give an explicit construction of a polynomial curve \cC(t) = (x(t), y (t)), where °x = 3, °y = N + 1 + 2\pent N4 that has exactly N crossing points \cC(t_i)= \cC(s_i) whose parameters satisfy s₁ < ... < s_N < t₁ < ... < t_N. Our proof makes use of the theory of Stieltjes series and Padé approximants. This allows us an explicit polynomial parametrization of the torus knot K_2,N.

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