The lexicographic degree of the first two-bridge knots (Q2214725)

This article studies polynomial parametrisations of knots and in particular the polynomial degrees that are in some sense minimal for a given knot type. Every knot \(K\) admits a polynomial parametrisation, i.e. there is a polynomial map \(\gamma:\mathbb{R}\to\mathbb{R}^3\) whose image closes to \(K\) in \(S^3\) by adding the point at infinity. To every such parametrisation we can associate a triple of numbers, namely the polynomial degrees of the parametrisation of the \(x\)-, \(y\)- and \(z\)-coordinate, respectively. The lexicographic degree of a knot \(K\) is defined as the triple that is minimal among all polynomial parametrisations of \(K\) with respect to the lexicographic order. The authors use techniques from the study of plane curves and pseudoholomorphic curves to compute the lexicographic degrees of all 2-bridge knots with minimal crossing number at most 11 and find that for these knots the lexicographic degree always takes the form \((3,b,3N-b)\) for some value \(b\).