Narrow Results

Let $K$ be a link of Conway’s normal form $C(m)$, $m\ge 0$, or $C(m,n)$ with $mn>0$, and let $D$ be a trigonal diagram of $K.$ We show that it is possible to transform $D$ into an alternating trigonal diagram, so that all intermediate diagrams remain trigonal, and the number of crossings never increases.

We study the degree of polynomial representations of knots. We obtain the lexicographic degree for two-bridge torus knots and generalized twist knots. The proof uses the braid theoretical method developed by Orevkov to study real plane curves, combined with previous results from [Chebyshev diagrams for two-bridge knots, Geom. Dedicata150 (2010) 405–425; E. Brugallé, P.-V. Koseleff, D. Pecker, Untangling trigonal diagrams, to appear in J. Knot Theory and its Ramifications]. We also give a sharp lower bound for the lexicographic degree of any knot, using real polynomial curves properties.