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arxiv: 1910.05045 · v2 · submitted 2019-10-11 · 🧮 math.GT · math.GR

Remarks on some maximal subgroups of F and on the vec{F}-index of knots

Valeriano Aiello This is my paper

Pith reviewed 2026-05-24 14:12 UTC · model grok-4.3

classification 🧮 math.GT math.GR
keywords Thompson group Fmaximal subgroupsstabilizer subgroupsvec F-indexknotsorientation reversalrectangular subgroupJones construction
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The pith

Three maximal subgroups of infinite index in K_{(2,2)} of Thompson's group F are stabilizer subgroups, and the vec F-index of a knot increases by at most 3 when its orientation is reversed.

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

The paper examines subgroups of the Thompson group F inside its rectangular subgroup K_{(2,2)}. It identifies three maximal subgroups of infinite index that each contain Jones's 3-colorable subgroup and shows these can be described as stabilizer subgroups. It also proves that the vec F-index, a knot invariant coming from Jones's Thompson-group construction, rises by no more than 3 when a knot's orientation is reversed. A reader would care because the characterizations give explicit descriptions of large subgroups while the bound supplies a practical limit on how the invariant behaves under reversal.

Core claim

We demonstrate that three maximal subgroups of infinite index in the rectangular subgroup K_{(2,2)} of the Thompson group F, each containing Jones's 3-colorable subgroup F, can be characterized as stabilizer subgroups. Additionally, we show that the vec F-index may increase at most by 3 after changing the orientation of a knot.

What carries the argument

Stabilizer subgroups inside the rectangular subgroup K_{(2,2)} of Thompson's group F that contain the 3-colorable subgroup and serve to identify the maximal ones; the vec F-index as the knot invariant derived from the group construction.

Load-bearing premise

The rectangular subgroup K_{(2,2)} and the vec F-index are defined exactly as in the earlier literature on Thompson groups and Jones's knot construction.

What would settle it

An explicit knot whose vec F-index increases by 4 or more upon orientation reversal would disprove the stated bound.

Figures

Figures reproduced from arXiv: 1910.05045 by Valeriano Aiello.

Figure 1
Figure 1. Figure 1: Pairs of opposing carets in F2 and F3 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The injection ι : F2 → F3 is induced by the following map between binary and ternary trees. 7→ group theoretical interpretation for the Thompson group F3 and, thus, providing an answer to Jones’s question [20, Question (4)]. Given an element (T+, T−) of F3, we introduce its Thompson permutation P(T+, T−), which is given by the composition of two permutations crafted from the two trees. Then, we prove that … view at source ↗
Figure 3
Figure 3. Figure 3: A element of F2, its image under the injection ι : F2 → F3, and an element in F3 \ ι(F2). Y = ι(Y ) = Z = by any pair (T, T) and the inverse of (T+, T−) is just (T−, T+). There is a natural injection ι : F2 ֒→ F3. Given (T+, T−) ∈ F2, firstly, add a new leaf to the middle of each vertex (thus turning every trivalent vertex into a quadrivalent one, see [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The rules needed to turn 4-valent vertices into crossings. 7→ 7→ [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Rules for calculating the permutation. We note that in every tree there exists exactly one path from a leaf, say f, to the root. For this path we consider the permutation (0, f). For example, in our case we have (1, 5), (2, 4), (0, 3), (4, 2), (5, 1). Since all the transpositions (but the one corresponding to the root) occur exactly twice, we set aside only one of each. Now we define the permutation π(T+) … view at source ↗
Figure 6
Figure 6. Figure 6: A graphical representation of the rules described in (2), (3), (4) of Proposition 1. a1 a2 a3 a4 a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 a7 a8 Before stating the main result of this paper, we provide a characterisation of tangled permutations. We recall that a permutation on {0, . . . , n} determines a partition consisting of its orbits. We say that a 4-tuple a < b < c < d is a crossing if {a, c} and {b, d} be… view at source ↗
read the original abstract

We demonstrate that three maximal subgroups of infinite index in the rectangular subgroup \( K_{(2,2)} \) of the Thompson group \( F \), each containing Jones's \( 3 \)-colorable subgroup \( \mathcal{F} \), can be characterized as stabilizer subgroups. Additionally, we show that the \( \vec{F} \)-index, an elementary knot invariant introduced thanks to Jones's construction of knots from Thompson groups, may increase at most by $3$ after changing the orientation of a knot.

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

0 major / 1 minor

Summary. The paper demonstrates that three maximal subgroups of infinite index in the rectangular subgroup K_{(2,2)} of Thompson's group F, each containing Jones's 3-colorable subgroup F, can be characterized as stabilizer subgroups. It additionally proves that the vec F-index of a knot increases by at most 3 under orientation reversal.

Significance. If the characterizations and bound hold, the work supplies explicit stabilizer descriptions for these subgroups inside K_{(2,2)} and a uniform orientation bound for the vec F-index, both derived directly from the Thompson-group knot construction; such explicit group-theoretic descriptions and elementary bounds on invariants are useful for further study of Jones-type knot invariants.

minor comments (1)
  1. The abstract and provided excerpt state the main results clearly, but without access to the full derivations, proofs, or explicit definitions of the three subgroups and the stabilizer maps, the technical details cannot be verified.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript and for the positive assessment of its significance. The recommendation is listed as uncertain, but the report contains no specific major comments or points requiring clarification. We therefore respond to the overall evaluation below and confirm that the characterizations and bound in the paper are as stated in the abstract.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives explicit characterizations of three maximal subgroups of K_{(2,2)} as stabilizers (each containing the 3-colorable subgroup) and proves a uniform +3 bound on the vec F-index change under orientation reversal. Both results follow directly from the Thompson-group knot construction and standard prior definitions of the rectangular subgroup and vec F-index, without any self-definitional reduction, fitted inputs renamed as predictions, or load-bearing self-citations that would force the claims by construction. The central statements remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

No free parameters or invented entities; the work relies on standard mathematical axioms and domain assumptions from the literature on Thompson groups and knot invariants.

axioms (2)
  • domain assumption Thompson group F and its rectangular subgroup K_{(2,2)} have the properties used in the characterizations.
    Invoked in the statement about maximal subgroups.
  • domain assumption The vec F-index is defined via Jones's construction from Thompson groups and behaves as an elementary knot invariant under orientation changes.
    Central to the second claim.

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Reference graph

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