{"arxiv_id":"2611.09063","title":"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 $q=e^{2\\pi i/N}$ 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.","primary_cat":"math.GT","cross_cats":[],"submitted_at":"2026-11-06T06:42:49+00:00","latest_version":1,"source":"arxiv","source_label":"arXiv","source_url":null,"has_pdf":true,"has_source":false,"verdict":"ACCEPT","confidence":"MODERATE","one_line_summary":"Cables of connected sums of figure-eight knots provide counterexamples to the full-sequence volume limit for knots.","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","upvotes":0,"downvotes":0,"pith_score":0,"inbound_count":0,"_links":{"html":"https://pith.science/paper/2611.09063","verdict":"/api/v1/verdicts/2611.09063"}}