The Crossing Number and Arc Index of Satellite Knots

Abstract: By considering arc presentations and grid diagrams of satellite knots, we prove that if K is a satellite knot with companion C and wrapping number w, then its crossing number satisfies $c(K) \geq \frac{2}{5}(w<sup>2c(C))<sup>{1/11}$. We also prove other similar bounds regarding the crossing number and arc index of satellite knots and links. This is partial progress on Problem 1.2 of the K3 Problem List.