On the degree of the colored Jones polynomial

Abstract

The extreme degrees of the colored Jones polynomial of any link are bounded in terms of concrete data from any link diagram. It is known that these bounds are sharp for semi-adequate diagrams. One of the goals of this paper is to show the converse; if the bounds are sharp then the diagram is semi-adequate. As a result, we use colored Jones link polynomials to extract an invariant that detects semi-adequate links and discuss some applications.

To appear in Acta Math. Vietnamica (Proceedings of Hyperbolic Geometry and Quantum Topology in Nha Trang)