Jones diameter and crossing number of knots

TL;DR

This paper characterizes when the quadratic degree bound of the colored Jones polynomial is sharp, linking it to adequacy, and applies this to determine crossing numbers of various complex knots.

Contribution

It establishes a precise condition for the sharpness of the degree bound in terms of adequacy and computes crossing numbers for broad classes of non-adequate prime satellite knots.

Findings

Bound is sharp iff the knot is adequate.

Determined crossing numbers for Whitehead doubles of adequate knots.

Computed crossing numbers for connected sums involving Whitehead doubles.

Abstract

It has long been known that the quadratic term in the degree of the colored Jones polynomial of a knot is bounded above in terms of the crossing number of the knot. We show that this bound is sharp if and only if the knot is adequate. As an application of our result we determine the crossing numbers of broad families of non-adequate prime satellite knots. More specifically, we exhibit minimal crossing number diagrams for untwisted Whitehead doubles of zero-writhe adequate knots. This allows us to determine the crossing number of untwisted Whitehead doubles of any amphicheiral adequate knot, including, for instance, the Whitehead doubles of the connected sum of any alternating knot with its mirror image. We also determine the crossing number of the connected sum of any adequate knot with an untwisted Whitehead double of a zero-writhe adequate knot.