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arxiv: 2407.05343 · v3 · submitted 2024-07-07 · 🧮 math.GT · math.DS

Studying knots in self-covers of the modular flow

Sivan Eldar , Stav Fahima This is my paper

Pith reviewed 2026-05-23 23:21 UTC · model grok-4.3

classification 🧮 math.GT math.DS
keywords modular knotsAnosov flowstrefoil complementself-coveringsGhys templateclosed geodesicscommensurable links
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The pith

Lifting Ghys' modular template via self-coverings of the trefoil complement gives combinatorial access to knot types of closed geodesics in the resulting Anosov flows.

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

The paper constructs templates for infinitely many Anosov flows on the trefoil complement by lifting Ghys' modular template through self-coverings of order 6k+1 for positive integers k. These lifted templates supply a combinatorial device for examining the knot properties of closed geodesics in the flows. The same construction produces an explicit infinite family of two-component links in which one component is the trefoil and all members are commensurable to one another.

Core claim

By lifting Ghys' modular template along self-coverings of the trefoil complement of order 6k+1, the authors obtain templates for the corresponding Anosov flows that preserve both the Anosov property and the combinatorial structure required for knot analysis, thereby permitting direct study of the knot types realized by closed geodesics and the explicit construction of an infinite commensurable family of two-component links with the trefoil as one component.

What carries the argument

The lifted versions of Ghys' modular template obtained from self-coverings of the trefoil complement of order 6k+1, which carry the preserved combinatorial data for tracking knot types in the Anosov flows.

Load-bearing premise

Self-coverings of the trefoil complement of order 6k+1 exist and admit lifts of Ghys' modular template that keep both the Anosov property and the combinatorial structure intact.

What would settle it

For any specific k, failure to produce a lift of the template that remains Anosov while preserving the combinatorial data needed to read off knot types from the template would refute the general construction.

Figures

Figures reproduced from arXiv: 2407.05343 by Sivan Eldar, Stav Fahima.

Figure 1.1
Figure 1.1. Figure 1.1: (a) The modular template; (b) One block of our [PITH_FULL_IMAGE:figures/full_fig_p002_1_1.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: 1: Fibration of the trefoil complement: (a) A regular fiber (blue) and the two [PITH_FULL_IMAGE:figures/full_fig_p003_2_1.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: 2: (a) Intersection points of a regular fiber [PITH_FULL_IMAGE:figures/full_fig_p004_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: 1: Farey tessellation of (a) The Poincar [PITH_FULL_IMAGE:figures/full_fig_p005_2_2.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: 2: The modular surface SMod Considering F, it is easily seen that when a geodesic crosses an ideal triangle in F and does not point to one of its ideal vertices, it produces an oriented segment that connects two edges of that triangle. These two edges share a vertex, so we label the segment by R if the vertex lies to the right of the segment, and L if the vertex lies to its left, as shown in figure 2.2.3… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: 5: The modular template after gluing the edges [PITH_FULL_IMAGE:figures/full_fig_p007_2_2.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: 6: The xy-loop We call passing through the template’s left ear an x-loop, and passing through its right ear a y-loop (see figure 2.2.7). Notice that the symbols x and y correspond exactly to R and L, respectively. The xy-loop is the concatenation of x and y [PITH_FULL_IMAGE:figures/full_fig_p008_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: 1: The once-punctured torus H Lemma 2.3.1 The action of φ corresponds to rotating H by an angle of 2π 6 counter￾clockwise. Proof. P SL2(Z) has two subgroups of order 6, one corresponds to a cover of the modular surface by the once-punctured torus, and one corresponds to a cover by the three-punctured sphere. These two covers imply from the two ways of gluing ideal triangles (which form an ideal square) i… view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: 2: The 0-disk, as a result of gluing the edges We also obtain the twisted bands α, β, γ by gluing every pair of opposite edges. In figure 2.3.3, every pair of regions of the same color represents one twisted band [PITH_FULL_IMAGE:figures/full_fig_p010_2_3.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: 5: H1 Proof. Notice that H1 := H × I, and in H1 the top and bottom faces are identified with each other and represent the return of Σ to itself after acting φ. The movement from the bottom face to the top face describes the sweeping of Σ in the 3-space (induced by the action on S 1 ). The interval I corresponds to φ acting once, which, as we saw earlier, is equivalent to rotating the top face by 2π 6 . ■… view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: 7: Meridian (blue) and longitude (orange), with orientation [PITH_FULL_IMAGE:figures/full_fig_p012_2_3.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: The modular template in H1 Proof. First, visualize the embedding of the modular template in the trefoil complement together with the xy-loop, as seen in figure 3.2. Considering the xy-loop in this context would be beneficial in §4 [PITH_FULL_IMAGE:figures/full_fig_p013_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: The modular template embedded in the trefoil complement [PITH_FULL_IMAGE:figures/full_fig_p013_3_2.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: The template’s boundary The red part’s composition of meridians and the branchline allows us to prove its embed￾ding in H1 is as depicted in the figure 3.4. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p014_3_3.png] view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: The embedding of the red boundary part in [PITH_FULL_IMAGE:figures/full_fig_p014_3_4.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: The modular template’s boundary in comparison to [PITH_FULL_IMAGE:figures/full_fig_p015_3_5.png] view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: The correspondence between the longitudes [PITH_FULL_IMAGE:figures/full_fig_p015_3_6.png] view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: Dividing the boundary into parts The following table organizes the different parts of the purple boundary’s path. We set our starting point to be p2, denoted by 1, and our ending point to be p1, denoted by 11. Step no. From point ... to point ... Description I 1 → 2 We travel along λ1 within the 0-disk until we reach twisted band α. II 2 → 3 We enter twisted band α with λ1, then we travel half a meridian… view at source ↗
Figure 3.8
Figure 3.8. Figure 3.8: The corresponding path in H1 In conclusion, the entire boundary of the modular template is composed of the path we have described and of the xy-loop, which are connected through p1 and p2 [PITH_FULL_IMAGE:figures/full_fig_p017_3_8.png] view at source ↗
Figure 3.9
Figure 3.9. Figure 3.9: The modular template’s boundary in H1 Now, we are ready to embed the template. By gluing identified edges and vertices, the path we described generates a one-component continuous surface. However, when describing the embedding, we will refer to four individual parts, each bounded by some continuous part of the boundary and by a face of H1. The first pair of individual parts we consider appears in figure … view at source ↗
Figure 3.10
Figure 3.10. Figure 3.10: The modular template with color correspondence to figures 3.11 and 3.12 [PITH_FULL_IMAGE:figures/full_fig_p018_3_10.png] view at source ↗
Figure 3.11
Figure 3.11. Figure 3.11: First pair of individual parts, from two angles [PITH_FULL_IMAGE:figures/full_fig_p018_3_11.png] view at source ↗
Figure 3.12
Figure 3.12. Figure 3.12: Second pair of individual parts, from two angles [PITH_FULL_IMAGE:figures/full_fig_p018_3_12.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: 1: The xxy-loop (the colors correspond) Along the curve drawn in figure 4.1.1 (b), some parts are directed upwards, and some are directed downwards. The upward-directed movement is consistent with the orientation we chose on the meridian in note 2.3.4. Thus, every y symbol corresponds to an upward movement along a meridian, while every x symbol corresponds to a downward movement. However, the lift of the… view at source ↗
read the original abstract

In this paper we provide a combinatorial tool to help study some topological properties of modular knots. We construct templates for the infinitely many Anosov flows on the trefoil complement, which are lifts of the geodesic flow on the modular surface, by lifting Ghys' modular template using self-covering of the trefoil complement of order $6k+1$, for $k\in \mathbb{N}_{>0}$. This allows to study the knot properties of closed geodesics in these flows, and an explicit construction of an infinite family of links of two components with one of them being the trefoil, all commensurable to one another.

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.

Referee Report

1 major / 0 minor

Summary. The paper constructs templates for infinitely many Anosov flows on the trefoil complement as lifts of Ghys' modular template via self-coverings of order 6k+1 (k>0). These templates are used to analyze knot properties of closed geodesics in the flows and to give an explicit infinite family of two-component links, all commensurable to one another, with one component the trefoil.

Significance. If the lifts are shown to preserve the Anosov property and the required combinatorial data, the work supplies a combinatorial tool for studying modular knots in self-covers and an explicit construction of an infinite commensurable family of links containing the trefoil.

major comments (1)
  1. [Construction via self-covers (following the abstract)] The central construction asserts that self-covers of the trefoil complement of order 6k+1 admit lifts of Ghys' modular template that remain Anosov and retain the branch structure needed for knot analysis. No explicit verification of preserved expansion/contraction rates, no reference to a general theorem guaranteeing hyperbolicity under these covers, and no check that the combinatorial data is unaltered for arbitrary k appear in the manuscript; this is load-bearing for all subsequent claims about closed geodesics and the infinite link family.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for explicit verification of the central construction. We address the major comment below.

read point-by-point responses

  1. Referee: The central construction asserts that self-covers of the trefoil complement of order 6k+1 admit lifts of Ghys' modular template that remain Anosov and retain the branch structure needed for knot analysis. No explicit verification of preserved expansion/contraction rates, no reference to a general theorem guaranteeing hyperbolicity under these covers, and no check that the combinatorial data is unaltered for arbitrary k appear in the manuscript; this is load-bearing for all subsequent claims about closed geodesics and the infinite link family.

    Authors: We agree that the manuscript lacks an explicit verification of these properties in the submitted version. The construction relies on the fact that covers of order 6k+1 are compatible with the modular flow's periodic data, but we did not supply a direct check of the expansion/contraction rates or a citation to a general result on preservation of the Anosov property under such lifts. In the revised manuscript we will insert a new subsection (or appendix) that (i) computes the effect of the lift on the hyperbolic metric and verifies that the expansion and contraction rates remain strictly greater than 1 in absolute value, (ii) references standard results on the stability of Anosov flows under finite covers when the cover degree preserves the relevant homology classes, and (iii) confirms by direct inspection of the template's branch lines that the combinatorial data (branching pattern and periodic orbit correspondence) is unchanged for every k>0. These additions will make the load-bearing claims fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity; construction extends external Ghys template via standard self-coverings

full rationale

The derivation constructs templates for Anosov flows on the trefoil complement by lifting Ghys' modular template through self-coverings of order 6k+1. This relies on an external reference (Ghys) and a standard covering-space operation rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation. No equation or claim reduces the target knot properties or link family to quantities defined in terms of themselves; the Anosov preservation and combinatorial structure are stated as assumptions to be checked against the external template, leaving the chain self-contained against independent benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence and liftability of Ghys' modular template together with the existence of self-covers of the stated orders; these are standard domain assumptions in 3-manifold dynamics rather than new postulates.

axioms (2)
  • domain assumption Ghys' modular template encodes the geodesic flow on the modular surface and lifts appropriately under self-covers of the trefoil complement.
    The construction begins by lifting this external template; its properties are taken as given.
  • domain assumption Self-coverings of the trefoil complement of degree 6k+1 exist for every positive integer k and preserve the Anosov character of the lifted flow.
    The paper invokes these covers to produce the infinite family of templates and links.

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 1 internal anchor

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