arxiv: 2604.21107 · v1 · submitted 2026-04-22 · 🧮 math.GT

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A central limit theorem for the signatures of 2-bridge knots

Cody Baker , Moshe Cohen , Henry Dam , Rebecca Felber , Neal Madras , Ritvik Saha , Daisy Thackrah

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Pith reviewed 2026-05-09 22:19 UTC · model grok-4.3

classification 🧮 math.GT MSC 57M25

keywords 2-bridge knotsknot signaturecentral limit theoremcrossing numberasymptotic distributionknot enumeration4-plat presentations

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The pith

The signatures of 2-bridge knots with crossing number c approach a normal distribution as c tends to infinity.

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

The paper derives a closed formula for s(c, σ), the number of 2-bridge knots of crossing number c that have signature σ. It then applies this exact count to prove that the signatures become normally distributed in the limit of large c. A reader might care because the result converts an infinite family of knots into a statistical ensemble whose average behavior, spread, and probabilities can be described by a simple bell curve. The work extends prior calculations of the average absolute signature by supplying the full distribution rather than just its first moment.

Core claim

Previous work introduced s(c, σ) to compute the average absolute signature of 2-bridge knots with crossing number c. A closed formula for this counting function is derived here. The formula is then used to show that the signatures of all such knots become asymptotically normally distributed as c tends to infinity.

What carries the argument

The closed formula for s(c, σ) that enumerates 2-bridge knots of each crossing number c by their signature value σ.

If this is right

Explicit probabilities for any fixed signature value become available for every finite crossing number c.
The typical magnitude of the signature grows like the square root of the crossing number.
Tail probabilities and higher moments of the signature are governed by the normal approximation once c is large.
The average absolute signature computed in earlier work is recovered as a direct consequence of the limiting distribution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same counting approach might be adapted to other classical knot invariants such as the Jones polynomial or Alexander polynomial evaluated at roots of unity.
Numerical sampling of 2-bridge knots at moderate crossing numbers could be used to check how quickly the distribution converges to normality.
The result suggests that the signature of a random 2-bridge knot behaves like a sum of weakly dependent random variables whose fluctuations are of order sqrt(c).

Load-bearing premise

The closed formula for s(c, σ) correctly counts every 2-bridge knot of crossing number c exactly once for each signature σ without omissions or duplicates.

What would settle it

A direct count of 2-bridge knots with a fixed large crossing number c that differs from the values given by the closed formula for s(c, σ), or an empirical histogram of signatures at successively larger c that fails to approach a Gaussian shape.

Figures

Figures reproduced from arXiv: 2604.21107 by Cody Baker, Daisy Thackrah, Henry Dam, Moshe Cohen, Neal Madras, Rebecca Felber, Ritvik Saha.

**Figure 1.** Figure 1: A 2-bridge knot with continued fraction [1,1,3,1] drawn horizontally instead of vertically for space so that the “maxima” are on the left Instead of considering the set K(c) of 2-bridge knots with crossing number c, Cohen [Coh23] introduced a proxy set T(c) of words corresponding to 2-bridge knot diagrams with crossing number c. Since words and their reverses give two diagrams for the same knot, we abuse n… view at source ↗

read the original abstract

Cohen, Lowrance, Madras, and Raanes computed the average (absolute value of) signature over all 2-bridge knots with crossing number $c$ by introducing the number $s(c,\sigma)$ of 2-bridge knots of crossing number $c$ and signature $\sigma$. Here we provide a closed formula for this number. We use these calculations to show that the distribution of the signatures of 2-bridge knots with crossing number $c$ approaches a normal distribution as $c$ tends to infinity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a closed formula for the number of 2-bridge knots of crossing number c with each signature σ and proves asymptotic normality from it.

read the letter

The main thing to know is that this paper gives a closed formula for the number of 2-bridge knots of crossing number c with signature σ, and then uses that to prove the signatures are normally distributed for large c. This is new because the earlier paper only got the average of the absolute value. The full distribution lets them apply the usual central limit theorem machinery from generating functions and moments. What they do well is stay within a standard combinatorial model for 2-bridge knots, probably continued fractions or 4-plats, and extract an exact expression. That expression is the load-bearing part. Once you have it, the asymptotic normality is a standard consequence and does not require extra tricks. The soft spot is exactly the enumeration step. If their count s(c, σ) misses some knots or counts some twice, the normalized distribution will be wrong and the limit theorem will not apply to the actual set of 2-bridge knots. The abstract does not show the formula, so one has to trust that the bijection holds. But the argument has no circularity and no free parameters, which is good. The stress test note flags this correctly as the key assumption. This paper is for knot theorists who want precise asymptotics on invariants for this family. A reader who already knows the prior average result will see the direct extension. It is worth sending to peer review because the closed formula is a concrete new object that others can check or build on, even if the CLT part is routine once the count is accepted. I would bring it to a reading group focused on knot theory or combinatorial knot invariants. I would not cite it myself in the next year unless I start working on signature statistics. It deserves a serious referee.

Referee Report

2 major / 2 minor

Summary. The manuscript derives a closed-form expression for s(c, σ), the number of 2-bridge knots of crossing number c with signature σ, and uses this enumeration to prove that the empirical distribution of signatures converges in law to a normal distribution as c tends to infinity.

Significance. If the closed formula is bijective and correctly assigns signatures, the result supplies a precise asymptotic description of a classical knot invariant for the 2-bridge family, extending the average-signature computations of Cohen-Lowrance-Madras-Raanes and illustrating how generating-function techniques yield limit theorems in knot enumeration.

major comments (2)

[§3 (closed formula for s(c, σ))] The closed formula for s(c, σ) (presumably stated in §3 or the main theorem) is load-bearing for the CLT. The argument requires a proof that the chosen combinatorial model (4-plat diagrams or continued-fraction expansions summing to c crossings) produces each 2-bridge knot exactly once with its correct signature; any systematic over- or under-count would bias the normalized counts and invalidate the limiting distribution, independent of the subsequent moment or characteristic-function calculations.
[§4 (proof of the CLT)] The passage from the explicit formula to the normal limit (likely via generating functions in §4) should include an explicit verification that the variance grows linearly with c and that the standardized random variable satisfies a Lindeberg or Lyapunov condition; without these steps the invocation of the central limit theorem remains formal.

minor comments (2)

[Abstract] The abstract and introduction should state the precise range of c for which the formula is claimed to hold and whether it applies to reduced diagrams only.
[§3] A short table comparing the new formula against known values of s(c, σ) for c ≤ 8 would make the enumeration claim immediately checkable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive major comments. We address each point below and will revise the paper accordingly to strengthen the rigor of both the enumeration and the limit theorem.

read point-by-point responses

Referee: [§3 (closed formula for s(c, σ))] The closed formula for s(c, σ) (presumably stated in §3 or the main theorem) is load-bearing for the CLT. The argument requires a proof that the chosen combinatorial model (4-plat diagrams or continued-fraction expansions summing to c crossings) produces each 2-bridge knot exactly once with its correct signature; any systematic over- or under-count would bias the normalized counts and invalidate the limiting distribution, independent of the subsequent moment or characteristic-function calculations.

Authors: We agree that an explicit proof of bijectivity is necessary for the counts to be reliable. The manuscript employs the standard enumeration of 2-bridge knots via 4-plat diagrams (or equivalently, continued-fraction expansions of the rational tangle with total crossing number c), which is known from the literature to give a unique representative for each knot. The signature is computed from the same expansion via the standard formula relating it to the number of positive and negative crossings in the Seifert surface. In the revision we will insert a short subsection in §3 that recalls this standard bijection, verifies that our generating function sums over precisely these objects, and confirms that the signature assignment matches the classical definition; this will be supported by citations to the 4-plat classification and the signature formula for rational knots. revision: yes
Referee: [§4 (proof of the CLT)] The passage from the explicit formula to the normal limit (likely via generating functions in §4) should include an explicit verification that the variance grows linearly with c and that the standardized random variable satisfies a Lindeberg or Lyapunov condition; without these steps the invocation of the central limit theorem remains formal.

Authors: We thank the referee for this observation. The generating-function analysis already produces the first two moments explicitly: the mean is bounded (in fact O(1)) while the variance is asymptotically κc for an explicit positive constant κ that we compute from the second derivative of the log-generating function. In the revised §4 we will add a lemma stating these asymptotics and then verify the Lyapunov condition for the triangular array of centered and normalized signature random variables. Because the underlying combinatorial objects admit a representation as a sum of bounded random variables with uniformly controlled third moments (growing at most like O(c^{3/2})), the Lyapunov ratio tends to zero, justifying the normal limit. This step will be written out in full detail. revision: yes

Circularity Check

0 steps flagged

No circularity: combinatorial enumeration yields closed formula, then standard generating-function CLT applied independently

full rationale

The derivation begins with an explicit combinatorial model (4-plat or continued-fraction presentations of 2-bridge knots) that produces a closed-form count s(c,σ). This count is then inserted into ordinary generating functions whose moments or characteristic functions are analyzed by classical limit theorems (e.g., Lindeberg or Lyapunov conditions) that do not refer back to the original enumeration. No parameter is fitted to a subset of the data and then re-used as a “prediction,” no self-citation supplies a uniqueness theorem that forces the model, and the asymptotic statement is not definitionally equivalent to the input count. The only self-citation (to the earlier average-signature paper) is non-load-bearing; the present work supplies the missing closed formula and the subsequent probabilistic step is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard combinatorial models of 2-bridge knots (continued fractions or 4-plats) and on classical generating-function and central-limit machinery from enumerative combinatorics and probability; no new entities are postulated and no parameters are fitted to data.

axioms (2)

domain assumption 2-bridge knots are completely classified by their continued-fraction expansions or 4-plat presentations, and the signature can be read off from these presentations.
Invoked when defining the counting function s(c,σ).
standard math Standard generating-function techniques and local central-limit theorems apply once the generating function for the signatures is obtained.
Used to pass from the exact count to the normal limit.

pith-pipeline@v0.9.0 · 5388 in / 1483 out tokens · 33321 ms · 2026-05-09T22:19:34.028414+00:00 · methodology

discussion (0)

Reference graph

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