Minimal Polynomial Parameterization of Rational Knots

Explore the mathematical theory of polynomial parameterizations of knots in this 45-minute research lecture. Delve into the concept of polynomial maps from real numbers to three-dimensional space whose closures in the three-sphere are isotopic to given knots. Learn about the lexicographic degree of knots, defined as the minimal degree under lexicographic ordering for polynomial parameterizations. Examine specific results for two-bridge knots with 12 or fewer crossings, including methods for estimating total degrees of lexicographic parameterizations. Discover how this problem connects to the study of real algebraic trigonal plane curves and the application of braid theoretical methods developed by Orevkov. Follow the collaborative research findings presented by Pierre-Vincent Koseleff from Institut de Mathématiques de Jussieu-Paris Rive Gauche, conducted jointly with E. Brugallé and D. Pecker, offering insights into the intersection of knot theory, algebraic geometry, and polynomial parameterization techniques.