Cable Knot Counterexamples to the All-Knot Volume Conjecture

Abstract

Let $E$ be the figure-eight knot and let $V$ be the hyperbolic volume of its exterior. We study the reduced colored Jones polynomial at $q=e^{2πi/N}$ for the $(p,w)$-cable of the connected sum of $d\ge 2$ copies of $E$, where $w\ge 2$ and $p,w$ are coprime. The normalized logarithmic growth rate converges to the JSJ volume $dV$ along colors coprime to $2w$, and its upper limit along colors sharing a divisor with $w$ is at most $V$. Thus the $(1,2)$-cable of $E\#E$ violates the full-sequence volume limit for knots. The $(1,3)$-cable of the same companion shows that convergence can fail even along all odd colors. The separation comes from the multiples of $N$ in the exact cabling sum and a uniform first-derivative estimate for the other companion colors.