The higher order terms in asymptotic expansion of color Jones polynomials

Color Jones polynomial is one of the most important quantum invariants in knot theory. Finding the geometric information from the color Jones polynomial is an interesting topic. In this paper, we study the general expansion of color Jones polynomial which includes the volume conjecture expansion and the Melvin-Morton-Rozansky (MMR) expansion as two special cases. Following the recent works on SL(2,C) Chern-Simons theory, we present an algorithm to calculate the higher order terms in general asymptotic expansion of color Jones polynomial from the view of A-polynomial and noncommutative A-polynomial. Moreover, we conjecture that the MMR expansion corresponding to the abelian branch of A-polynomial. Lastly, we give some examples to illustrate how to calculate the higher order terms. These results support our conjecture.

💡 Research Summary

The paper investigates the full asymptotic expansion of the colored Jones polynomial (J_n(K;q)) for a knot (K), unifying two previously separate expansions: the exponential growth term appearing in the Volume Conjecture and the power‑series expansion known as the Melvin‑Morton‑Rozansky (MMR) expansion. The authors start from the classical (A)-polynomial (A_K(L,M)=0), which encodes the character variety of the knot complement, and quantize it to obtain a non‑commutative operator (\hat A_K(\hat L,\hat M;q)) satisfying (\hat L\hat M = q,\hat M\hat L). This quantum (A)-polynomial annihilates the colored Jones polynomial, i.e. (\hat A_K(\hat L,\hat M;q),J_n(K;q)=0), thereby providing a difference equation that governs the entire sequence ({J_n}_{n\ge0}).

A central insight is that the zero set of the classical (A)-polynomial decomposes into distinct branches (or “branches”) in the ((L,M)) plane. The authors identify two relevant branches: (1) a non‑abelian branch, whose points correspond to non‑trivial flat (SL(2,\mathbb C)) connections. Expanding around this branch reproduces the exponential term of the Volume Conjecture together with an infinite tower of sub‑leading corrections. (2) an abelian branch, located near the trivial point ((L,M)=(1,1)). Expanding around this branch yields a formal power series in (\hbar=2\pi i/n) that matches precisely the MMR expansion. The paper conjectures that the MMR series is exactly the asymptotic expansion associated with the abelian branch of the (A)-polynomial.

To compute higher‑order terms systematically, the authors propose an algorithm consisting of four steps:

1. Obtain the classical and quantum (A)-polynomials for the knot under study. For many knots (e.g., figure‑eight, torus knots) these are already known; otherwise they can be derived from the gluing equations of an ideal triangulation.
2. Translate (\hat A_K) into a difference operator acting on the index (n). The operators (\hat L) and (\hat M) act as (\hat L,J_n = J_{n+1}) and (\hat M,J_n = q^{n/2} J_n), respectively.
3. Choose a branch and set initial data. For the non‑abelian branch one uses the leading exponential term dictated by the hyperbolic volume; for the abelian branch one uses the known values (J_0=1) and the first few low‑(n) polynomials.
4. Apply a WKB‑type expansion in the small parameter (\hbar). Substituting a formal series (\log J_n \sim \sum_{k\ge0} S_k \hbar^{k-1}) into the difference equation yields recursive relations for the coefficients (S_k). Solving these recursions produces the full set of higher‑order corrections.

The authors demonstrate the method on several examples. For the figure‑eight knot they recover the known volume term (\frac{\text{Vol}(4_1)}{2\pi}) and compute the first few sub‑leading corrections, which agree with numerical data. In the abelian branch they reproduce the MMR coefficients up to order (\hbar^4) and show exact agreement with the original MMR formula. Similar calculations for torus knots (T(2,2p+1)) and the knot (5_2) further support the conjecture that the MMR series corresponds to the abelian branch.

The paper’s contributions are twofold. First, it provides a conceptual bridge linking the exponential growth captured by the Volume Conjecture and the perturbative series captured by MMR through a single geometric object, the (A)-polynomial and its quantization. Second, it supplies a practical computational tool that can be applied to any knot for which the quantum (A)-polynomial is known, enabling the systematic extraction of arbitrarily high‑order terms in the asymptotic expansion of the colored Jones polynomial.

In the discussion, the authors outline several directions for future work. One is the extension of the algorithm to other quantum invariants such as the HOMFLY‑PT polynomial or the Kauffman polynomial, which have their own quantum (A)-polynomials. Another is the exploration of higher‑rank gauge groups (e.g., (SL(N,\mathbb C))) and the associated multi‑variable (A)-polynomials, potentially revealing richer branch structures. Finally, they suggest that the interplay between the quantum (A)-polynomial and three‑dimensional Chern‑Simons theory could shed light on the quantization of character varieties and on the relationship between knot invariants and quantum gravity. Overall, the work establishes a unified framework for understanding the full asymptotic behavior of colored Jones polynomials and opens new avenues for both mathematical and physical investigations.