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arxiv: 2509.21274 · v2 · submitted 2025-09-25 · 🧮 math.GT

On the arc index and Turaev genus of a link

\'Alvaro Del Valle V\'ilchez , Adam M. Lowrance This is my paper

Pith reviewed 2026-05-18 14:06 UTC · model grok-4.3

classification 🧮 math.GT
keywords arc indexTuraev genuscrossing numberadequate linkspositive 3-braidsalternating linkstorus linksKanenobu knots
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The pith

Adequate links have their arc index computed exactly, with bounds established for positive 3-braid closures and a conjecture on an inequality with crossing number and Turaev genus verified across multiple families.

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

The paper computes the arc index of an adequate link. It also establishes bounds on the arc index of the closure of a positive 3-braid. The authors conjecture an inequality between the crossing number, arc index, and Turaev genus of a link. They show that this conjecture holds for several infinite families, including alternating links, links with Turaev genus one, adequate links, closures of positive 3-braids, torus links, and most Kanenobu knots. If correct, the conjecture would allow one to relate and bound these three link invariants in a new way.

Core claim

We compute the arc index of an adequate link and establish bounds on the arc index of the closure of a positive 3-braid. We also conjecture an inequality between the crossing number, arc index, and Turaev genus of a link and show the conjecture is true for several infinite families of links including alternating links, links with Turaev genus one, adequate links, closures of positive 3-braids, torus links, and most Kanenobu knots.

What carries the argument

The conjectured inequality between crossing number, arc index, and Turaev genus, supported by direct computations on adequate diagrams and positive 3-braid diagrams.

If this is right

  • Adequate links now have an exact formula for their arc index in terms of crossing number.
  • All closures of positive 3-braids have explicit upper and lower bounds on arc index.
  • The inequality holds for every alternating link and every torus link.
  • The inequality holds for most Kanenobu knots, extending the relation to many non-alternating examples.

Where Pith is reading between the lines

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

  • If the conjecture holds for all links, Turaev genus would supply a systematic upper bound on arc index once crossing number is known.
  • The same diagram techniques used here could be applied to test the inequality on additional families such as pretzel links.
  • The arc index bounds for 3-braids might combine with existing braid index results to constrain the geometry of these links.

Load-bearing premise

The computations and verifications rest on the standard definitions and basic properties of adequate links, positive 3-braids, crossing number, arc index, and Turaev genus as previously established in the knot theory literature.

What would settle it

A concrete link outside the verified families, such as a specific Kanenobu knot not covered by the 'most' cases or a non-torus link with Turaev genus two, where the conjectured inequality between crossing number, arc index, and Turaev genus fails.

Figures

Figures reproduced from arXiv: 2509.21274 by Adam M. Lowrance, \'Alvaro Del Valle V\'ilchez.

Figure 1
Figure 1. Figure 1: Four different ways of representing the trefoil as an arc presentation. using only Cromwell moves that preserve or decrease the number of pages, yielding an algorithm to detect the unknot. Because the arc index of a link is defined as a minimum, its computation often relies on a lower bound. Beltrami and Morton [63] proved that the difference between the maximum and minimum a-degree of the two-variable Kau… view at source ↗
Figure 2
Figure 2. Figure 2: The Legendrian front diagrams FD and FD∗ obtained from the grid diagram in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: On the left, a spanning tree T of a link diagram D is indicated by bold edges. On the right, the portion of D in a regular neighborhood of T is blue, while the rest of D is red. The dotted curve η is the boundary of a regular neighborhood of T. Assign a height to each arc on the interior of η as follows. Arbitrarily pick any arc and arbitrarily assign it a height. Now iteratively choose an arc γ that cross… view at source ↗
Figure 4
Figure 4. Figure 4: A planar isotopy transforms the diagram from [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Isotopies resulting in unnested exterior arcs. Since our goal is to minimize the number of arcs or spokes, we hope to find a spanning tree with many reducible edges that results in few exterior arcs like those in the bottom-right of [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The diagram on the left is obtained from the diagram in [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The spoke diagram obtained from the diagram on the right in [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: In both pictures, a partial filtered spanning tree is indicated by bold edges. On the left, the edge f is bad. On the right, the dotted curve is a cutting arc. Jin and Lee [31] proved that, given a filtered spanning tree T, the nested exterior arcs - such as those in the bottom right of [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The edge ei of the filtered spanning tree is labeled with i. Edges marked with a small disk are reducible edges. 2.2. Turaev genus. Following the discovery of the Jones polynomial [35], Kauffman [42], Murasugi [65], and Thistlethwaite [73] independently proved several of Tait’s conjectures about alternating knots. A key element in each of their proofs is the inequality span VL(t) ≤ c(L) where equality hold… view at source ↗
Figure 10
Figure 10. Figure 10: The cobordism between the all-A and all-B states of D is a saddle in a neighborhood of each crossing of D. genus. Bae [4] also used alternating tangle decompositions to get bounds on arc index. Bae’s work on alternating decompositions applies to a proper subset of adequate links. Turaev genus has been computed for several infinite families of links. Non-alternating pretzel and Montesinos links have Turaev… view at source ↗
Figure 11
Figure 11. Figure 11: An adequate, Turaev genus two diagram that does not contain a span￾ning tree with four reducible edges. 4. Closed positive 3-braids The main goal of this section is to prove Conjecture 1.1 for a link that is the closure of a positive 3-braid. 4.1. Garside normal forms. The braid group on n strands, Bn, has the following presentation Bn = ⟨σ1, . . . , σn−1 ∣ σiσj = σjσi , ∣i − j∣ > 1 σiσjσi = σjσiσj , ∣i −… view at source ↗
Figure 12
Figure 12. Figure 12: Diagrams D and D′ (of β = ∆pσ k1 1 , depending on the parity of p), their all-B states, and their B-Lando graphs [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Generic diagram of β̂, with β ∈ Λ4 (a) and β ∈ Λ5 (b), their all-B states and their B-Lando graphs. The next theorem gives bounds on the arc index of the closure of a positive 3-braid. Theorem 4.10. Let β be a 3-braid in Λi, for some i ∈ {1, 2, 3, 4, 5}. (1) Let β = ∆p ∈ Λ1. Then α(β̂) = 6 if p = 0, p + 4 ≤ α(β̂) ≤ 1 2 (3p + 1) + 3 if p > 0 and odd, p + 4 ≤ α(β̂) ≤ 1 2 (3p) + 3 if p > 0 and even [PITH_FU… view at source ↗
Figure 14
Figure 14. Figure 14: Generic diagram of β̂ and a good spanning tree shaded on it, with β = ∆p ∈ Λ1. Edges marked with a small disk are reducible edges. Remark 4.11. If γ ∈ Λi , i ∈ {1, 2, 3, 4, 5}, its infimum is explicitly described by the representatives given in Proposition 4.3. In addition, it turns out that inf s(γ) = inf(γ) (see, e.g., [23, Rem. 2.3]). Consequently, Theorem 4.10 explicitly calculates the arc index for t… view at source ↗
Figure 15
Figure 15. Figure 15: Generic diagram of β̂ and a good spanning tree shaded on it, with β = ∆pσ k1 1 ∈ Λ2. Edges marked with a small disk are reducible edges [PITH_FULL_IMAGE:figures/full_fig_p023_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Generic diagram of β̂ and a good spanning tree shaded on it, with β = ∆2uσ1σ2 ∈ Λ3. Edges marked with a small disk are reducible edges. We are now ready to prove Conjecture 1.1 for closed positive 3-braids [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Generic diagrams of β̂′ (a) and β̂ (b), and a good spanning tree shaded on them, with β ′ = ∆2uσ k1−1 1 σ k2 2 ⋯σ k2t 2 σ1 and β = ∆2u+1σ k1 1 σ k2 2 ⋯σ k2t+1 1 ∈ Λ5. Edges marked with a small disk are reducible edges. Theorem 4.14. Let β be a positive 3-braid such that β̂ is prime and non-split. Then, c(β̂) + 2 − α(β̂) ≥ 2gT (β̂), and thus Conjecture 1.1 is true for closed positive 3-braids. Proof. Accor… view at source ↗
Figure 18
Figure 18. Figure 18: p and q odd [PITH_FULL_IMAGE:figures/full_fig_p029_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The Kanenobu knot K(p, q). The rectangle labeled n contains a twist region with n crossings, as indicated. have Turaev genus one, the conjecture holds. Now suppose that pq ≥ 3. Qazaqzeh and Mansour [72] proved that c(K(p, q)) = p + q + 8, and Lee and Takioka [50] proved that α(K(p, q)) = p + q + 6. Therefore, c(K(p, q)) + 2 − α(K(p, q)) = (p + q + 8) + 2 − (p + q + 6) = 4 ≥ 2gT (K(p, q)). Suppose that pq … view at source ↗
read the original abstract

We compute the arc index of an adequate link and establish bounds on the arc index of the closure of a positive 3-braid. We also conjecture an inequality between the crossing number, arc index, and Turaev genus of a link and show the conjecture is true for several infinite families of links including alternating links, links with Turaev genus one, adequate links, closures of positive 3-braids, torus links, and most Kanenobu knots.

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 / 3 minor

Summary. The paper computes the arc index of adequate links and establishes bounds on the arc index of the closure of a positive 3-braid. It conjectures an inequality relating the crossing number cr(L), arc index α(L), and Turaev genus g_T(L) of a link, and verifies the conjecture for several infinite families including alternating links, links with Turaev genus one, adequate links, closures of positive 3-braids, torus links, and most Kanenobu knots.

Significance. If the results hold, the explicit computation of arc index for adequate links and the bounds for positive 3-braid closures provide concrete advances in evaluating these invariants. The verification of the conjectured inequality cr(L) ≥ α(L) − 2g_T(L) across multiple infinite families supplies substantial supporting evidence and strengthens the case for the relation. These contributions build directly on standard definitions of adequate diagrams, Turaev surfaces, and arc presentations already in the literature.

minor comments (3)
  1. The abstract states the conjecture but does not give its precise form (e.g., cr(L) ≥ α(L) − 2g_T(L)); including the exact inequality would improve clarity for readers.
  2. In the section presenting the arc-index computation for adequate links, the argument would benefit from an explicit statement of which known properties of adequate diagrams are invoked at each step.
  3. For the verification on Kanenobu knots, a brief table or list indicating which specific knots are covered and which are excluded would make the scope of the result easier to assess.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately reflects the main results, including the computation of arc index for adequate links, bounds for positive 3-braid closures, and verification of the conjectured inequality across multiple infinite families.

Circularity Check

0 steps flagged

No significant circularity; derivations rest on independent literature definitions

full rationale

The paper computes the arc index of adequate links and bounds for positive 3-braid closures directly from the standard definitions of adequate diagrams, Turaev surfaces, and arc presentations already established in the knot theory literature. The conjecture cr(L) ≥ α(L) − 2g_T(L) is verified on infinite families (alternating links, Turaev genus one links, adequate links, torus links, etc.) by invoking known equalities or inequalities for those classes rather than fitting parameters or reducing to self-citations. All load-bearing steps are self-contained against external benchmarks and do not reduce by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on prior established definitions and properties of link invariants in knot theory; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math Standard definitions and properties of link invariants including crossing number, arc index, Turaev genus, adequate links, and positive 3-braids
    These background concepts are invoked to state the computations, bounds, and conjecture verifications.

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