Appears in collection : Combinatorics and Arithmetic for Physics

A polynomial parameterization of a knot $K$ in $S^3$ is a polynomial map $\gamma : \mathbf{R} \to \mathbf{R}^3$ whose closure of the image in $S^3$ is isotopic to $K$. Every knot admits a polynomial parametrisation, and we are interested in determining the lexicographic degree of a knot $K$, i.e. the minimal degree for the lexicographic order of a polynomial parametrisation of $K$. We give the lexicographic degree of all two-bridge knots with 12 or fewer crossings. First, we estimate the total degree of a lexicographic parametrisation of such a knot. This allows us to transform this problem into a study of real algebraic trigonal plane curves, and in particular to use the braid theoretical method developed by Orevkov. Joint work with E. Brugall\'e an D. Pecker