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The Ozsv\'ath-Szab\'o tau-invariant of braided satellites

T0 review · 1 major / 1 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read Braided satellite knots have explicit tau and epsilon formulas, and none with winding number at least two induce concordance homomorphisms.

desk verdict The paper gives explicit tau and epsilon formulas for braided satellite knots by defining squeezed patterns and shows none with winding number at least 2 induce concordance homomorphisms, but the formula rests on an unverified transfer from the knot case. read the letter →

arxiv 2606.10351 v1 pith:F2WEINZ4 submitted 2026-06-09 math.GT

Alex Eldridge This is my paper · ORCID
classification math.GT
keywords tau-invariantbraidedsatellitessqueezedpatternsknotconcordancesatelliteknotsepsiloninvariantwindingnumberHeegaardFloerhomology
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 derives formulas for the tau and epsilon concordance invariants of satellite knots whose patterns are braided. It introduces squeezed patterns by direct analogy with squeezed knots and proves every braided pattern is squeezed, which supplies a tau formula for the entire class. From this it follows that no squeezed pattern, hence no braided pattern, with absolute winding number two or more can induce a group homomorphism on the knot concordance group. A reader would care because the formulas make many satellite invariants computable and the homomorphism result restricts the algebraic maps that satellite operations can produce.

What carries the argument

Squeezed patterns, the class of solid-torus patterns that admit a tau formula by the same mechanism used for squeezed knots.

What would settle it

An explicit braided pattern with winding number two whose induced map on concordance classes is a non-zero homomorphism would refute the central claim.

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Extended reading notes

Core claim

The author proves that the tau invariant of any satellite formed by a squeezed pattern equals a concrete expression in the pattern and companion invariants, that every braided pattern is squeezed, and therefore that no squeezed or braided pattern with absolute winding number at least two defines a homomorphism from the concordance group to the integers.

Load-bearing premise

The tau formula known for squeezed knots extends without change to the definition of squeezed patterns inside the solid torus.

Editorial extensions

If this is right

  • Tau and epsilon become computable for every braided satellite knot.
  • No braided pattern with absolute winding number two or greater can induce a homomorphism on the concordance group.
  • The result applies to the larger class of all squeezed patterns, not only braided ones.

Reading between the lines

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

  • The same obstruction may apply to other concordance invariants such as upsilon.
  • Patterns with winding number exactly one remain the only plausible candidates for producing homomorphisms.
  • Direct computation on low-crossing braided examples could confirm or bound the formulas.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 1 minor

Summary. The manuscript provides formulas for the Ozsváth-Szabó τ-invariant and the ε-invariant of satellite knots whose patterns are braided. It introduces the class of squeezed patterns by direct analogy with squeezed knots of Feller-Lewark-Lobb, proves that every braided pattern is squeezed, states a τ formula for all squeezed patterns, and concludes that no squeezed (hence no braided) pattern with winding number at least 2 induces a homomorphism on the concordance group, addressing a conjecture of Hedden.

Significance. If the central formulas and the transfer of properties hold, the results would supply explicit, computable expressions for concordance invariants of a broad class of satellites and give partial progress on Hedden's conjecture. The introduction of squeezed patterns as an intermediate class is a natural and potentially reusable extension of prior work.

major comments (1)
  1. The τ formula for squeezed patterns (the extension of the Feller-Lewark-Lobb construction) is load-bearing for both the satellite formulas and the non-homomorphism claim. The manuscript defines squeezed patterns by analogy, proves braided patterns belong to the class, and states the formula, but supplies no independent Heegaard Floer computation or explicit verification on a non-trivial braided pattern confirming that the relevant filtration levels or correction-term vanishings transfer verbatim to the solid-torus pattern setting.
minor comments (1)
  1. The introduction could state the explicit τ and ε formulas for braided satellites at the outset rather than deferring them entirely to later sections.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the load-bearing nature of the squeezed-patterns construction. We respond to the single major comment below.

read point-by-point responses

  1. Referee: The τ formula for squeezed patterns (the extension of the Feller-Lewark-Lobb construction) is load-bearing for both the satellite formulas and the non-homomorphism claim. The manuscript defines squeezed patterns by analogy, proves braided patterns belong to the class, and states the formula, but supplies no independent Heegaard Floer computation or explicit verification on a non-trivial braided pattern confirming that the relevant filtration levels or correction-term vanishings transfer verbatim to the solid-torus pattern setting.

    Authors: The definition of squeezed patterns is deliberately formulated so that the Heegaard-diagram simplifications and filtration arguments of Feller-Lewark-Lobb apply verbatim once the pattern is placed in the solid torus; the proof that every braided pattern is squeezed consists precisely in exhibiting, for an arbitrary braided pattern, a diagram in which the same generator filtrations and differential vanishings hold. Consequently the τ formula follows by the identical correction-term computation. We agree, however, that an explicit numerical check on a non-trivial braided pattern would make the transfer more transparent and will include such a verification (together with the corresponding filtered chain complex) in the revised manuscript. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivation relies on new definitions and external analogy without reduction to inputs

full rationale

The provided abstract and context describe defining squeezed patterns analogously to Feller-Lewark-Lobb (distinct prior authors), proving braided patterns belong to this class, and stating a τ formula for squeezed patterns. No equations, self-citations, or load-bearing steps are quoted that reduce the formula or homomorphism result to a fitted parameter, self-defined quantity, or unverified carry-over by construction. The chain is presented as independent extension and proof, qualifying as self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Review is abstract-only; no free parameters, axioms, or invented entities can be extracted beyond the new definition of squeezed patterns, which is a definitional class rather than a postulated physical entity.

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Cite this review

Pith. "Pith review of The Ozsv\'ath-Szab\'o tau-invariant of braided satellites." pith.science (2026). https://pith.science/paper/F2WEINZ4

@misc{pith2026260610351,
  author       = {Pith},
  title        = {Pith review of: The Ozsv\'ath-Szab\'o tau-invariant of braided satellites},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F2WEINZ4}},
  note         = {Machine review of arXiv:2606.10351}
}
abstract

We give formulas for the $ \tau $ and $ \varepsilon $ concordance invariants of satellite knots whose patterns are braided, meaning they wind around the solid torus without reversing. Our methods lead us to define the class of squeezed patterns, analogous to squeezed knots as defined by Feller-Lewark-Lobb. We show that all braided patterns are squeezed, and we give a $ \tau $ formula for squeezed patterns as well. Towards a conjecture of Hedden, we show that no squeezed pattern, and thus no braided pattern, with winding number at least 2 induces a homomorphism on the concordance group.

Figures

Figures reproduced from arXiv: 2606.10351 by the authors.

Figure 1
Figure 1. A braid β which closes to a knot and its correponding pattern Pβ, the closure of β in the solid torus. Theorem 1.1. For an index-p braid β whose closure is a knot and a knot K ⊂ S 3 , we have τ (Pβ(K)) =    pτ (K) − 1 2 (p − 1) + 1 2w(β) if ε(K) = 1 pτ (K) + 1 2 (p − 1) + 1 2w(β) if ε(K) = −1 τ (Pβ(U)) if ε(K) = 0, where w(β) is the writhe of β. 1 arXiv:2606.10351v1 [math.GT] 9 Jun 2026 [PITH_FULL_IMAGE:figure… view at source ↗
Figure 2
Figure 2. Three patterns and their induced maps on C, which are conjectured to be the only homomorphisms induced by satellites. Much partial progress has been made in proving this conjecture. In [LMP24], it was shown that patterns satisfying a technical condition on linking numbers in branched covers do not induce homomorphisms, and in [JKM25], this condition was shown to hold for all patterns with a winding number that is ev… view at source ↗
Figure 3
Figure 3. The construction of a genus-(p − 1)q cobordism from Cp,−q to Cp,q of which an arbitrary braid is a slice [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Left: a braid β with a Seifert-framed longitude marked in blue. Right: the braid β2,q representing the (2, q)-cable of the pattern Pβ [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: A pattern P and its corresponding slice pattern P #. Or, in other words, the satellite P #(K) is P(K) # −P(U). So, when P is squeezed with lower squeezing cobordism Σ− : Cp,−q → P, its corresponding slice pattern P # has the following partial τ formula for K with ε(K) …
Figure 6
Figure 6. Figure 6: Four patterns corresponding to small crossing number links and their representations as braids with two strands running around the solid torus and one strand not. To save space, patterns are drawn in D2 × [0, 1] with the top and bottom edges identified. Example 5.1. In…
Figure 7
Figure 7. Figure 7: Squeezing cobordisms for the patterns in [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: A construction of a non-braided squeezed pattern. Example 5.3. To construct a non-braided squeezed pattern, we can return to the cobordisms discussed in the proof of Theorem 1.1, those from Cp,−q to Cp,q con￾structed by attaching a band resolving each negative crossing…

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Reference graph

Works this paper leans on

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