Knot Cabling and the Degree of the Colored Jones Polynomial

Abstract

We study the behavior of the degree of the colored Jones polynomial and the boundary slopes of knots under the operation of cabling. We show that, under certain hypothesis on this degree, if a knot $K$ satisfies the Slope Conjecture then a $(p,q)$-cable of $K$ satisfies the conjecture, provided that $p/q$ is not a Jones slope of $K$. As an application we prove the Slope Conjecture for iterated cables of adequate knots and for iterated torus knots. Furthermore we show that, for these knots, the degree of the colored Jones polynomial also determines the topology of a surface that satisfies the Slope Conjecture. We also state a conjecture suggesting a topological interpretation of the linear terms of the degree of the colored Jones polynomial (Conjecture \ref{conj}), and we prove it for the following classes of knots:iterated torus knots and iterated cables of adequate knots, iterated cables of several non-alternating knots with up to nine crossings, pretzel knots of type $(-2,3,p)$ and their cables, and two-fusion knots.

31pages. Modified to also include the results of arXiv:1501.04614. This version will appear in the New York Journal of Mathematics

References in corpus (6)

• Slopes and colored Jones polynomials of adequate knots
• A cabling formula for the colored Jones polynomial
• Quadratic integer programming and the slope conjecture
• The AJ-conjecture and cabled knots over the figure eight knot
• The slope conjecture for graph knots
• The AJ-conjecture and cabled knots over torus knots

Cited by in corpus (2)

• The slope conjecture for graph knots
• Crossing numbers of cable knots