Cable Knot Counterexamples to the All-Knot Volume Conjecture

Haibin Hang has a nice result. The construction is straightforward once you see it: take a connected sum of figure-eight knots (d≥2), cable it with winding number w≥2, and use the arithmetic of the Morton formula to separate the growth rate along different color subsequences. The JSJ volume is dV, but along colors sharing a factor with w the normalized invariant stays below V, so the limit cannot exist. The concrete examples in Corollary 2 (C_1,2(E#E) and C_1,3(E#E)) are crisp and the failure is strict.

The paper is well-organized. The main theorem is stated precisely, and the derivation follows from three ingredients: (i) the Habiro–Lê formula for the figure-eight invariants, (ii) the Morton cabling formula, and (iii) a lemma bounding |J_m(E;ζ_N)| and |D J_m(E;ζ_N)| for m not a multiple of N. The first two are standard; the third is the technical core.

Where is the soft spot? Lemma 3. The value bound is fine (short, clean). The derivative bound is sketched via a Temperley–Lieb trace argument, and the algebra is not written out in full. I believe the constant C_B exists, but the derivation jumps from matrix element estimates to a sum of O(N⁴ T_N) and O(N³ T_N) terms without checking cancellations in the product–derivative interplay. A referee will want a more explicit computation, especially for the range r ≤ ℓ < N−r where the zero factor is differentiated. The argument is plausible and matches known results, but it is not yet a finished proof.

That said, the rest of the paper is rigorous. The counting of multiples of N in the cabling sum and the comparison of the main term to H_N are clear and quantitative. The exponential decay of the error term relies on d≥2, which is the essential novelty.

Who is this for? Knot theorists and anyone interested in the volume conjecture. The paper is a genuine step: it shows that the full-sequence limit fails for satellites, not just for links. It does not disprove the conjecture for hyperbolic knots, but it sharpens the boundaries.

Bottom line: the result is correct in spirit and likely correct in detail. The trace argument needs expansion, but that is fixable. This paper deserves peer review at a good topology journal.

If correct, the results provide the first counterexamples to the full-sequence all-knot volume conjecture for knots (as opposed to links). The examples are explicit and the argument is self-contained modulo known results. The paper sharpens the understanding of how subsequence limits can differ from the JSJ volume, which is of interest to quantum topology and the volume conjecture program.

This paper constructs counterexamples to the all-knot volume conjecture, which asserts that the normalized logarithmic growth rate of the colored Jones polynomial at root-of-unity converges to the JSJ volume of the knot exterior for every knot. By studying the (p,w)-cable of a connected sum of two or more figure-eight knots, the author proves that along some arithmetic progressions of colors the limit equals twice the JSJ volume, while along others it stays bounded by the volume of a single figure-eight exterior. This separation implies that the full integer sequence does not converge, violating the conjecture. The result relies on an exact cabling sum, a uniform first-derivative estimate for the figure-eight invariants, and the exponential decay of error terms when the number of summands is at least two.

What carries the argument. The central objects are: (1) the reduced colored Jones polynomial J_N(K;q) evaluated at q = $e^{{2π i/N}}$; (2) the Habiro-Lê formula for the figure-eight knot E, which gives J_m(E;q) as a sum of products of brace factors; (3) the Morton–van der Veen cabling formula for the (p,w)-cable of a connected sum, expressing J_N(L;q) as a sum weighted by quantum dimensions. The load-bearing pieces are Lemma 3, which provides the value bound |J_m(E;ζ_N)| ≤ $N^{2}$ and the derivative bound |DJ_m(E;ζ_N)| ≤ C_B $N^{5}$ T_N for m not divisible by N, and the observation that when d ≥ 2 the error term H_N is exponentially smaller than the dominant term T_N^d after division by [N].

Core claim. The central discovery is that for the cable knot L = C_{1,2}(E#E) (where E is the figure-eight knot), the limit of R_N(L) along odd N is 2V, while its upper limit along even N is at most V, so the full-sequence limit fails. For L = C_{1,3}(E#E), the limit along N ≡ 1 mod 6 is 2V and the upper limit along N ≡ 3 mod 6 is at most V, so even the limit over all odd colors fails. More generally, for any d ≥ 2 and w ≥ 2, the limit along colors coprime to 2w equals the JSJ volume dV, while the upper limit along colors sharing a divisor with w is at most V. The JSJ volume of the cable is still attained as the upper limit over all N, but the lower limit is strictly smaller due to the sub-sequence gaps.

The premise it rests on. The uniform derivative bound |DJ_m(E;ζ_N)| ≤ C_B $N^{5}$ T_N for m not a multiple of N (Lemma 3) relies on a Temperley-Libe trace argument that is only sketched; the existence of a constant C_B independent of N and valid for all m ≤ BN may depend on cancellations that are not fully verified for the worst-case braid word.

The all-knot volume conjecture is disproved for infinite families of cable knots built from composite figure-eight companions.

For any cable of E#d with d ≥ 2, the full-color limit fails to be a single number; the limit exists only along certain arithmetic subsequences.

The JSJ volume of the cable knot is still recovered as the limit superior over all colors, but the limit inferior is strictly smaller, showing that the conjecture cannot be salvaged by replacing limit with limit superior.

The arithmetic selection of colors (coprime to 2w vs having common divisor with w) is necessary and determined by the winding number w and the parity of the cable parameters.

Similar arithmetic splitting likely occurs for cables of other hyperbolic knots with simple colored Jones asymptotics, provided the companion has a known derivative bound analogous to Lemma 3.

The failure of convergence along all odd colors for w=3 suggests that more refined subsequence dependence may appear for other winding numbers, possibly related to the structure of the cabling sum's arithmetic lattice.

If the derivative bound in Lemma 3 could be strengthened (e.g., to polynomial rather than quasi-exponential in N), the argument might extend to d=1, giving counterexamples from a single figure-eight companion—but the current reliance on d≥2 is essential for the exponential error decay.

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