Narrow Results

In this paper, we construct an algorithm to produce canonical grid diagrams of cable links and Whitehead doubles, which give sharper upper bounds of the arc index of them. Moreover, we determine the arc index of $2$-cable links and Whitehead doubles of all prime knots with up to eight crossings.

We prove some estimates on the crossing number of satellites of adequate, and partly of semiadequate knots. We mostly deal with classical cables, obtaining the first examples where the asymptotic cable crossing number is shown to be equal to the companion crossing number. Our approach can also be applied to Whitehead and Bing doubles and positive braid cables. For some patterns one can use it to determine their geometric degree.