Quantum Mechanics of Particle on a torus knot: Curvature and Torsion Effects

Dripto Biswas; Subir Ghosh

REVIEW 2 major objections 4 minor 54 references

Quantum Mechanics of Particle on a torus knot: Curvature and Torsion Effects

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A particle confined to a torus knot feels the knot's winding number in its energy spectrum.

desk verdict Worth a look for the GIP curvature/torsion computation on a torus knot, but the central cancellation identity is false and the advertised spectrum is unsupported. read the letter →

arxiv 1908.06423 v4 pith:5FFHV62Z submitted 2019-08-18 hep-th math-phmath.MPquant-ph

Dripto Biswas , Subir Ghosh This is my paper · ORCID

classification hep-thmath-phmath.MPquant-ph

keywords torusknotgeometry-inducedpotentialcurvaturetorsionwindingnumberMathieuequationconstrainedquantumdynamicstopologicalinvariant

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies a quantum particle constrained to move on a $(p,q)$ torus knot, a closed curve wound $p$ times around a torus and $q$ times around its tube. Using the geometry-induced potential approach, it argues that the curvature and torsion generated by the confinement change the effective Schrödinger equation, and it identifies a $\varphi$-independent combination $\Gamma$ built from the winding number $\alpha=q/p$ and the torus aspect ratio $b=\cosh(\eta_0)$, which it calls a topological invariant. In the thin-torus limit, with the torsion term neglected, the paper derives the energy eigenvalues $E_n=\hbar^2 n^2\cosh^2(\eta_0)/[2ma^2p^2(\alpha^2+\sinh^2(\eta_0)-1)]$, showing that the spectrum depends on both $p$ and $q$, unlike earlier results that saw only $p$. The paper also reports numerical comparisons indicating that its curvature-corrected energy formula tracks the simulated eigenvalues better than the earlier formula for larger $\alpha$.

What carries the argument

The central object is the effective Hamiltonian of the geometry-induced potential approach, $\hat{H}_{\mathrm{eff}}=-(\hbar^2/2m)(\nabla_s^2+\kappa^2/4-\tau^2/2)$, applied to the torus-knot embedding. The derivation reparametrizes arc length to the angle $\varphi$ through $d\varphi/ds=1/(a\beta\sigma^2)$, writes the wavefunction as $\psi=\sigma(\varphi)G(\varphi)$ with $\sigma(\varphi)=1/\sqrt{b-\cos(\alpha\varphi)}$, and relies on the identity that the metric term plus the curvature term equals the constant $\Gamma$ in (26)-(27), eliminating all $\varphi$-dependence from the geometric part of the equation. That reduction produces the Hill equation, and in the thin-torus limit the Mathieu equation (31); periodic boundary conditions fix the fractional order $\nu=2n/q$ and lead to the energy eigenvalues (33).

What would settle it

Substitute the paper's own expressions (19) for $\kappa^2/4$ and (25) for $(\sigma\sigma''-2\sigma'^2)/\sigma^2$ into the left side of (26) with $\alpha=3/2$ and $b=2$. At $\cos(\alpha\varphi)=0$ the sum is about $0.338$ and at $\cos(\alpha\varphi)=1$ about $0.097$, while the claimed $\Gamma$ is about $0.384$; these three numbers disagree, so the identity fails for this parameter choice.

Watch

Extended reading notes

Core claim

The paper's central claim is that a spinless particle constrained to move on a torus knot acquires a geometry-induced potential—specifically the $\kappa^2/4$ curvature term (and, before it is dropped, the $-\tau^2/2$ torsion term)—which earlier treatments of a particle on a knot omitted. In the large-winding-number, thin-torus regime the torsion term is numerically subdominant, and the time-independent Schrödinger equation reduces to a Mathieu equation whose coefficients contain $\alpha=q/p$. Enforcing $2\pi p$ periodicity selects Mathieu functions of fractional order $\nu=2n/q$ and yields the energy formula (33), so the knot's shape enters the spectrum through both $p$ and $q$. A second claim is that the geometric terms combine into the $\varphi$-independent constant $\Gamma=(b^2+\alpha^4-1)/(4(b^2+\alpha^2-1))$, which the authors call a topological invariant. If these claims are correct, knottedness is not erased by taking the thin-torus limit.

Load-bearing premise

The derivation stands on one algebraic identity—the angle-dependent geometric terms in the Schrödinger equation cancel to a constant independent of $\varphi$—and if that identity fails, the Mathieu reduction and the energy formula do not follow.

Editorial extensions

If this is right

Energy levels of a particle on a $(p,q)$ torus knot depend on both winding numbers $p$ and $q$ through $\alpha=q/p$, even in the thin-torus limit.
For large $\alpha$, the energy formula reduces to $E_n=\hbar^2 n^2\cosh^2(\eta_0)/(2ma^2q^2)$, so the poloidal winding number $q$ takes over from $p$.
For small $\alpha$ with $\alpha^2\ll\cosh^2(\eta_0)$, the result degenerates to the familiar particle-on-a-ring spectrum, recovering the known limit.
The full wavefunctions are Mathieu functions of fractional order $\nu=2n/q$ multiplied by $\sigma(\varphi)$, with $2\pi p$ periodicity imposed by the boundary conditions.
If the constant $\Gamma$ is genuinely independent of $\varphi$, it provides a new invariant combining the knot's winding number with the host torus's aspect ratio.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The identity (26) is the step where the whole reduction lives; if direct substitution of the paper's own curvature and metric formulas fails for some parameters, the Mathieu equation and the closed-form energy formula would not follow, and the numerical comparison in Section 6 should be checked against the exact equation (28) rather than the reduced one.
The name 'topological invariant' is attached to a quantity shown to be constant in $\varphi$ for one parametrization; showing that $\Gamma$ is unchanged by reparametrization or ambient isotopy would be a natural next step, and the paper does not provide that proof.
Because the torsion term is dropped on numerical grounds, an extension that keeps $-\tau^2/2$ could test whether the $\alpha$-dependence of the spectrum survives for small winding numbers, where torsion dominates curvature.
The same parameterization strategy could be applied to other curves with toroidal coordinate expressions, such as torus links or helical wires, to see whether an analogous constant $\Gamma$ appears.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the paper applies the standard da Costa–Wang geometry-induced potential to a torus knot and derives a candidate spectrum that depends on the knot type through α. That is a reasonable thing to try, and the curvature/torsion computation is fine as far as it goes. But the one algebraic step that makes the calculation tractable, Eq. (26), is wrong, and the main result collapses.

What is actually new: Sreedhar’s earlier work quantized a particle on a knot without the GIP curvature term. This paper recognizes that the GIP adds κ²/4 (and τ²/2, dropped in the large-α regime) and attempts to solve the resulting equation. The explicit formulas for κ and τ in the torus-knot parametrization, Eqs. (19) and (22), are not in Sreedhar, and the comparison of curvature vs torsion in Section 3.4 is useful. I see nothing wrong with the GIP framework section.

The soft spots are in the step that matters. Eq. (26) states that the sum of the metric term (25) and the curvature term a²β²σ⁴κ²/4 is a φ-independent constant Γ. It is not. Substituting the paper’s own Eqs. (19) and (25) for α=3/2, b=2 gives about 0.338 at cos αφ=0 and about 0.097 at cos αφ=1, while Γ≈0.384. The sum is visibly φ-dependent. That error is load-bearing: it converts Eq. (24) into the constant-coefficient Eq. (28), and it is what permits the Mathieu reduction and the energy formula (33). Without it, the Hill equation has more terms and the spectrum is not derived. The advertised ‘topological invariant’ is an algebraic artifact.

Eq. (31) also has a mistranscribed Mathieu coefficient: the change of variable z=αφ/2 introduces a factor 4/α² and the expansion gives the opposite sign for the cos(2z) term. The stated truncation order is also inconsistent with keeping that term. The numerical section is a six-point fit with an ad hoc linear term and no error bars, so it would not validate anything even if the algebra were right.

Who should read this? Someone working on GIP for curved quantum wires might skim Section 3. As a research claim, it is not usable. I would not cite it. If this crossed my desk, I would desk-reject with a note asking the authors to check Eq. (26); no referee time needed.

Referee Report

2 major / 4 minor

Summary. This paper studies the quantum mechanics of a spinless particle constrained to move on a (p,q) torus knot embedded in R^3. The authors use the geometry-induced potential (GIP) approach, starting from an effective Hamiltonian with curvature and torsion terms, present explicit expressions for the curvature and torsion of their knot parameterization, and argue that for large winding number α=q/p on a thin torus the torsion term is negligible. They then derive a one-dimensional Schrödinger equation for the angular coordinate and claim that the metric and curvature contributions add up to a φ-independent constant Γ depending only on b=cosh(η0) and α. On this basis they reduce the equation to Mathieu's equation in the thin-torus limit and obtain the energy eigenvalues E_n = ℏ^2 n^2 cosh^2(η0)/(2 m a^2 p^2(α^2+sinh^2(η0)−1)), which depends on both p and q through α. A short numerical comparison using six eigenvalues is presented as confirmation. The central claim of the paper is the φ-independence asserted in Eq. (26).

Significance. The topic is of genuine interest: curvature-induced potentials for constrained quantum systems are physically relevant, and a clean analytic spectrum for a torus knot would be a useful contribution. The manuscript is largely self-contained and has the merit of writing its key formulas explicitly, so the algebraic claims can be checked directly. The use of the established GIP effective Hamiltonian is appropriate, and the intended comparison with Sreedhar's results for a particle on a knot defines a clear benchmark. However, the main result rests on a single algebraic cancellation that, on substitution of the paper's own expressions, does not hold. Since Eq. (26) is the step that converts a variable-coefficient equation into a constant-coefficient one, the subsequent Hill/Mathieu treatment and the spectrum (33) are not established. The advertised 'topological invariant' Γ is therefore an algebraic artifact rather than a demonstrated invariant. The research question remains plausible, but the submitted version does not provide a valid derivation of its central claim.

major comments (2)

[Section 4, Eq. (26)] The claimed cancellation identity is false. Using the paper's own expressions (19) and (25), with x=cos(αφ), D=b^2+α^2−1, and A=(α^2−1)(2bx−x^2)+b^2+α^4−α^2, the left-hand side of Eq. (26) equals [α^2 D^2 (1−2bx+x^2) + (D−1)(b^2−1)A] / [4D^2 (b−x)^2], which is not independent of x. For example, take α=3/2 and b=2: the right-hand side is Γ=43/112≈0.3839, whereas the left-hand side is approximately 0.3376 at x=0 and approximately 0.0965 at x=1. Since Eqs. (28), (29), (31), and (33) all follow from the asserted φ-independence, the constant-coefficient reduction and the thin-torus spectrum are unsupported.
[Section 5.1, Eqs. (29)–(31)] Independently of the cancellation error, the binomial expansion and the change of variable contain further errors. Since σ^4=(b−cos(αφ))^{-2}=b^{-2}(1−cos(αφ)/b)^{-2}=b^{-2} Σ_{k≥0} (k+1)(cos(αφ)/b)^k, there is no (−1)^k factor in Eq. (29). Moreover, with the substitution 2z=αφ, one has d^2/dφ^2=(α^2/4)d^2/dz^2, so the coefficient of cos(2z) in Eq. (31) should contain an additional factor 4/α^2 relative to the expression written. The Mathieu equation stated in (31) is therefore not the thin-torus limit of (28), even if Eq. (26) were correct.

minor comments (4)

[Section 5.1, Eqs. (32)–(33)] The step from Eq. (32) to Eq. (33) drops an n-independent term while shifting the zero of energy. This is legitimate for level spacings, but the absolute eigenvalues of Eq. (31) contain that constant, and the numerical comparison in Section 6 appears to use absolute energies; the distinction should be stated explicitly.
[Section 4, Eq. (27)] The name 'topological invariant' for Γ is not substantiated. The quantity depends on the geometric parameters b and α, and no invariance under continuous deformations of the knot or the torus is demonstrated.
[Section 6, Eq. (42) and Fig. 6] A quadratic fit with an ad hoc linear term and only six data points, without error bars or residuals, is insufficient to support the quantitative claim of agreement between the analytic and numerical spectra; the authors should at least report fit residuals and the fitted parameters.
[Section 5.1, Eqs. (37)–(38)] The correction term written as ℏ^2/(16 m b p^2) inside the Mathieu function expansions has dimensions and is not consistent with an expansion in powers of 1/b; it should be checked and corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central identity (26) is an algebraic assertion with no fitted parameters, and all load-bearing inputs are external citations or direct substitutions.

full rationale

The derivation chain is self-contained in the relevant sense. The effective Hamiltonian (14) is imported from external prior work by da Costa [8] and Wang et al. [9,10], not from the present authors. The torus-knot parametrization (15) comes from an external paper [44]. The curvature and torsion expressions (19) and (22) are obtained by direct substitution into standard differential-geometry formulas. The pivotal step is Eq. (26), which asserts that the sum of the metric term (25) and the curvature term equals the constant Γ of Eq. (27). This is an algebraic identity claim, not a fitted parameter and not a quantity defined in terms of the target energy spectrum. Γ is an explicit function of the knot parameter α and torus parameter b, and it is not adjusted to reproduce the energy eigenvalues. The energy formula (33) follows from the Mathieu-equation analysis of (31) using an external stability condition (32) from Ince/McLachlan. The self-citations [53,54] appear only in the conclusion as plans for future work and play no load-bearing role in the derivation. The numerical fit (42) is a display of validation data, not a fitted parameter later renamed as a prediction. The reader's concern that Eq. (26) may fail numerically is a correctness issue, not a circularity issue: a false algebraic premise invalidates a derivation, but it does not make the derivation circular. Under the stated rules, circularity requires exhibiting a specific reduction of a claimed prediction to its own input, by construction or by fitted-parameter renaming, and no such reduction is present here.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

The central derivation rests on the imported da Costa/Wang Hamiltonian, the decision to drop torsion, the thin-torus expansion, and a false algebraic identity. No external data are used.

free parameters (1)

B (linear term in numerical fit) = Not stated; fitted to six numerical eigenvalues
Added in Eq. (42) to accommodate the lowest eigenvalue not being the ground state; no independent justification.

assumptions (5)

domain assumption Effective Hamiltonian H_eff = -(ℏ^2/2m)(Delta_s + κ^2/4 - τ^2/2), imported from da Costa and Wang
Adopted in Section 2, Eq. (14); assumes an infinitely strong confining potential and a normal-plane ground state.
domain assumption Torsion term can be neglected for large alpha on a thin torus
Stated in Section 3.4 and used from Section 4 onward based on a numerical comparison, not an analytic bound.
domain assumption Thin-torus approximation: b approximately c, with O(1/b^2) terms neglected
Used in Section 5.1, Eq. (31), to truncate the binomial expansion and obtain a Mathieu equation.
domain assumption Periodic boundary conditions G(0)=G(2πp) and G'(0)=G'(2πp)
Given in Eqs. (40) and (41); these conditions select the allowed Mathieu characteristic exponent and hence the energy quantization.
ad hoc to paper The sum in Eq. (26) is a constant Gamma independent of phi
Asserted without a complete derivation; direct substitution of Eqs. (19) and (25) shows the sum is phi-dependent for alpha=3/2, b=2.

invented entities (1)

Topological invariant Gamma
purpose: Claimed constant characterizing the torus-knot spectrum
Defined in Eq. (27); not shown to be invariant under knot isotopy, and the identity defining it fails when checked numerically.

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Cite this review

Pith. "Pith review of Quantum Mechanics of Particle on a torus knot: Curvature and Torsion Effects." pith.science (2026). https://pith.science/paper/5FFHV62Z

@misc{pith2026190806423,
  author       = {Pith},
  title        = {Pith review of: Quantum Mechanics of Particle on a torus knot: Curvature and Torsion Effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5FFHV62Z}},
  note         = {Machine review of arXiv:1908.06423}
}

read the original abstract

Constraints play an important role in dynamical systems. However, the subtle effect of constraints in quantum mechanics is not very well studied. In the present work we concentrate on the quantum dynamics of a point particle moving on a non-trivial torus knot. We explicitly take into account the role of curvature and torsion, generated by the constraints that keep the particle on the knot. We exploit the "Geometry Induced Potential (GIP) approach" to construct the Schrodinger equation for the dynamical system, obtaining thereby new results in terms of particle energy eigenvalues and eigenfunctions. We compare our results with existing literature that completely ignored the contributions of curvature and torsion. In particular, we explicitly show how the "knottedness" of the path influences the results. In the process we have revealed a (possibly un-noticed) "topological invariant".

Figures

Figures reproduced from arXiv: 1908.06423 by the authors.

**Figure 2.** (Left) A trefoil knot, or a (2, 3)−torus knot, is the simplest non-trivial knot. (Right) A (3,10)-torus knot. 3.1 Parametrisation of torus knots in toroidal coordinates Henceforth, we shall use toroidal coordinates, to exploit the inherent symmetry of the problem. We consider a torus of major radius R and minor radius d (see [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** (Left) The green surface represents τ(α,φ) 2 2 , while the yellow surface depicts κ(α,φ) 2 4 . (Right) The same plot, at a slightly different viewing-angle reveals the ’low α’ region, where torsion dominates curvature. The opposite happens for large α with α ≈ b providing roughly the demarcation line [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: (Left) The green curve represents an angular slice of the torsion surface plot in Fig. 3, corresponding to [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: (Left) Front-View of surface plot, with α ∈ (0, 1000) and η ∈ (3, 10). (Right) Top-View of the same plot. In the red zone F ≈ 1 where our results roughly agree with [44] whereas in the blue region the results greatly disagree. 5.2 Brief discussion on the results obtain…

**Figure 6.** Figure 6: (Left) The plot for α = 1.5. (Right) The plot for α = 3.5. In the left panel both results, ours (red curve) and [44] (blue curve) agree with the simulated result (green curve) whereas in the right panel, for larger α, the blue line [44] deviates significantly from the …

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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