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Analytic quantum invariants

This paper proposes a framework using deformed Borel-Laplace resummation to repackaging quantum invariants into analytic invariants, demonstrating the approach through explicit $(q,X)$ -series expressions for colored Jones polynomials of simple knots, Links-Gould polynomials of the figure-eight knot, and $q$ -series related to three-manifold invariants and mock modular forms.

Original authors: Rinat Kashaev, Vladimir Mangazeev, Marcos Marino

Published 2026-10-20

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CC BY 4.0

Original authors: Rinat Kashaev, Vladimir Mangazeev, Marcos Marino

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the hidden geometry of knots, mathematicians have long sought a way to see the invisible. A knot is not just a tangled string; it is a shape that can be described by a special kind of mathematical code, a polynomial that acts like a unique fingerprint. For decades, scientists have used these fingerprints to tell knots apart, but these codes only work for whole numbers, like counting the number of twists in a rope. They break down when you try to look at the space between those numbers, leaving a gap in our understanding of the knot's true, continuous nature. The challenge has been to turn these discrete, step-like codes into smooth, flowing functions that exist everywhere, allowing us to explore the knot as a whole object rather than a collection of separate points. This quest is vital because these knots are not just abstract puzzles; they are deeply connected to the structure of space itself and the behavior of quantum particles.

A team of researchers has now built a new bridge across this gap. They developed a method to take these broken, step-like codes and stitch them together into a single, smooth mathematical object that works for any value, not just whole numbers. They call this new object an "analytic invariant," a term that simply means a property that stays the same no matter how you look at it, but which now exists as a continuous function. To do this, they used a sophisticated technique called deformed Borel–Laplace resummation. Imagine trying to listen to a radio signal that is so full of static and gaps that it makes no sense; this method is like a powerful filter that cleans up the noise, fills in the missing pieces, and reveals the clear, continuous song underneath. By applying this filter to the mathematical codes of several famous knots, the team was able to create a complete, smooth picture of each knot's geometry.

The researchers tested their new method on some of the most famous knots in mathematics, including the figure-eight knot and the trefoil knot. For each one, they started with the known, step-by-step codes and applied their smoothing technique. The result was a new, continuous formula that perfectly matched the old codes at the whole numbers but also provided a clear answer for every value in between. This is a significant achievement because previous attempts to create such smooth formulas often failed or produced results that were singular, or broken, at specific points. The new formulas are robust and well-behaved, offering a reliable way to study the knot's properties across its entire range. The team found that these new smooth formulas are closely related to older, well-known mathematical series, but they are distinct. They contain extra, tiny corrections that were missing before, and these corrections are essential for the formulas to behave correctly when pushed to their limits.

The study also looked beyond simple knots to the shapes of three-dimensional spaces, known as manifolds, which can be created by cutting and rejoining knots. In this field, there are existing mathematical objects called invariants that describe these spaces, but they often struggle to connect with the smooth, continuous descriptions used in physics. The researchers applied their new smoothing technique to these space-invariants as well. They discovered that their method produces new, smooth functions that match the known discrete data but also fix a major problem: the old functions often failed to reproduce the correct behavior when the space was examined very closely. The new, smoothed-out versions successfully captured the correct behavior, revealing a deeper connection between the discrete data and the continuous geometry of these spaces.

One of the most striking findings was that these new smooth functions are not just simple versions of the old ones; they are more complex and richer. They include additional terms that act like corrections, ensuring that the math works out perfectly in every situation. These corrections are so small that they are often invisible in standard calculations, but they are crucial for the theory to hold together. The researchers showed that without these corrections, the mathematical description would fail to match the known physical properties of the knots and spaces. This suggests that the universe of these mathematical objects is more intricate than previously thought, with hidden layers of detail that only appear when you look with the right tools.

The paper does not claim to have solved every problem in the field. The researchers acknowledge that their method works for the specific knots and spaces they tested, but it is not yet known if it will work for every possible knot. They also note that their new functions are currently tied to the specific way the knots or spaces are presented, and it remains to be seen if the functions are truly independent of that presentation. However, the results are a strong step forward. By turning broken, step-like codes into smooth, continuous functions, the team has provided a new lens through which to view the geometry of knots and the topology of space. They have shown that with the right mathematical tools, the gaps in our understanding can be filled, revealing a continuous and coherent picture of the quantum world.

Technical Summary of "Analytic Quantum Invariants"

Problem Statement
The paper addresses the challenge of promoting quantum knot invariants, specifically the coloured Jones polynomials $J_{K,n}(q)$ , into analytic functions of continuous variables $q$ and $x$ (where $x$ interpolates the integer colour $n$ ). While the Melvin–Morton–Rozansky (MMR) series provides a formal power series expansion in $h = -\log q$ with coefficients depending on $X = qx$, this series is typically factorially divergent. Conventional Borel resummation techniques can convert this series into analytic functions, but these often involve "dual" variables ( $\tilde{q} = e^{-4\pi^2/h}$ ) and may not yield a single analytic function that is regular at integer values of the colour. Furthermore, existing proposals, such as the "inverted Habiro series," often fail to be analytic at positive integer values of the colour or do not asymptote correctly to the MMR series near $h=0$ . The authors seek a framework to define analytic invariants that are distinct from, yet closely related to, these existing constructions, providing a "non-perturbative completion" of the perturbative MMR series.

Methodology
The authors propose a framework based on deformed Borel–Laplace resummation. This approach utilizes specific $q$ -transforms defined for $0 < q < 1$ :

$q$ -Borel Transform: Defined as a Hadamard product with a specific kernel, converting a divergent series into a function with a finite radius of convergence.
$q$ -Laplace Transform: Defined as a genuine Lebesgue integral over the positive real half-line, distinct from standard $q$ -Laplace transforms used in linear $q$ -difference equations.
Iterative Resummation: For certain knots where a single resummation yields a divergent expression, the authors apply the procedure iteratively (e.g., twice for the $5_2$ knot).

The core of the method involves starting from Habiro's cyclotomic expansion of the coloured Jones polynomials:
$J_{K,n}(q) = \sum_{m=0}^{n-1} \frac{(q^{1+n}, q^{1-n}; q)_m}{\gamma_q(-m)} H_{K,m}(q)$
This is interpolated to a formal series $J_{K,x}(q, z)$ for non-integer $x$ . The authors apply the deformed Borel–Laplace resummation to this series. A key technical tool is the development of a general formalism for the resummation of basic hypergeometric series ( ${}_r\phi_s$ ) with $r > s+1$ , utilizing the Hadamard product of simple meromorphic functions (sm-functions) and residue calculus.

Key Contributions and Results

Definition of Analytic Jones Invariants: The authors define an analytic Jones invariant $\hat{J}_{K,x}(q)$ as the deformed Borel–Laplace resummation of the interpolating series. They prove that for $0 < q < 1$ , this invariant is analytic in $x$ (in suitable domains) and regular at $x=0$ . It specializes to the $|n|$ -coloured Jones polynomial at integer $x=n$ .
Explicit $(q, X)$ -Series Representations: The paper derives explicit expressions for these invariants in terms of $(q, X)$ $(q, X)$ -series (series in $q$ $q$ with Laurent polynomial coefficients in $X$ $X$ ).
- Figure-Eight Knot ( $4_1$ ): The authors provide a detailed analysis, deriving a contour integral representation and explicit formulas involving $q$ -Bessel functions and mock modular forms. They show the invariant is the sum of the inverted Habiro series and a "correction" term necessary to cancel singularities at $X=q^n$ and ensure correct asymptotics.
- Trefoil Knots ( $3_1^\pm$ ): Similar explicit formulas are derived, showing the invariant agrees with the real part of the Gukov–Manolescu invariant (proposed in [GM21]) but includes an imaginary correction term dependent on the branch choice of the logarithm.
- Knot $5_2$ : This case requires a double resummation (iterative application of the deformed Borel–Laplace procedure). The authors derive a complex expression involving constant terms of Laurent series and contour integrals, demonstrating the feasibility of the method for knots with more complex Habiro polynomials.
Links–Gould Polynomials: The framework is applied to a sequence of two-variable polynomials conjectured to be the Links–Gould polynomials for the figure-eight knot (associated with $U_q(\mathfrak{gl}(n|1))$ ), successfully interpolating them to non-integer values.
Three-Manifold Invariants (WRT): The method is extended to the Witten–Reshetikhin–Turaev (WRT) invariants of three-manifolds obtained via surgery. By applying the resummation to the $q$ $q$ -series arising from surgery formulas, the authors construct analytic functions whose formal asymptotic expansions reproduce the Ohtsuki series.
- Brieskorn Spheres: For manifolds like $\Sigma(2,3,7)$ and $\Sigma(2,3,5)$ , the resulting analytic functions are shown to differ from the proposed $\hat{Z}$ -invariants (which are often false theta functions or mock modular forms) by non-trivial correction terms. These corrections are essential for the functions to possess the correct asymptotic expansion (Ohtsuki series) as $h \to 0^+$ .
- Hyperbolic Manifold: The method is applied to a hyperbolic closed three-manifold ( $M_{-1/2}$ ), yielding an analytic function that corrects the known $\hat{Z}$ -invariant to match the perturbative expansion.

Significance and Claims
The paper claims to provide a systematic method for repackaging quantum invariants as analytic invariants that are distinct from conventional Borel resummations and inverted Habiro series.

Distinction from Existing Work: The authors emphasize that their invariants differ from conventional Borel resummations by exponentially small terms in $h$ (non-perturbative corrections) and do not generally depend on dual variables $\tilde{q}$ . They also differ from inverted Habiro series by being regular at positive integer colours and having the correct asymptotic behavior near $X=1$ .
Correction of $\hat{Z}$ -invariants: A significant finding is that the analytic functions derived via this method act as "corrected" versions of the $\hat{Z}$ -invariants found in the literature. The corrections are necessary to ensure the functions have the correct Ohtsuki asymptotic expansion, a property the raw $\hat{Z}$ -invariants often lack (e.g., lacking radial limits at roots of unity).
Modesty on Universality: The authors explicitly state that the method's universality for all knots is an open problem. They note that different knots may require different deformation parameters ( $p = q^\beta$ ) and that the procedure for the $5_2$ knot required an iterative approach, suggesting a need for further characterization of which knots require such iterations.
Surgery Dependence: The authors acknowledge that the analytic functions for three-manifolds are currently defined via specific surgery presentations. Invariance under changes of surgery presentation for $0 < q < 1$ has not been established and remains a major open problem.

In summary, the paper establishes a rigorous analytic framework for quantum invariants using deformed Borel–Laplace resummation, providing explicit formulas for several knots and three-manifolds, and revealing a structural relationship between these analytic invariants, mock modular forms, and the perturbative expansions of quantum field theories.

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