{"id":"48cd09d7-ea9b-4fe0-9785-6b13d08be6fb","arxiv_id":"2606.10351","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Formulas for tau and epsilon of braided satellite knots plus proof that no squeezed or braided pattern with winding number >=2 induces a concordance homomorphism.","lead":"The paper gives explicit formulas for the tau and epsilon concordance invariants of satellite knots with braided patterns and proves that none with winding number at least 2 induce homomorphisms on the knot concordance group. A smart generalist might read it to see how new classes of patterns help map the structure of the concordance group, which organizes knots up to 4-dimensional equivalence.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"The τ formula for squeezed patterns rests on an unverified carry-over of Feller-Lewark-Lobb properties to the satellite pattern setting.","rationale":"The reader’s weakest assumption correctly isolates the single point at which the entire chain of claims (formulas plus non-homomorphism) can fail. Because the manuscript supplies no machine-checked verification or independent HF calculation that would close this gap, the verdict must remain conditional until that check is performed.","tokens_in":1652,"tokens_out":351,"duration_ms":14310,"concrete_test":"Pick the simplest braided pattern with winding number 2 (e.g., the (2,3)-torus pattern or the standard positive braid pattern on two strands), form its satellite with the unknot and with trefoil, compute τ directly from the knot Floer complex of the resulting knots, and check whether the values match the paper’s claimed formula; disagreement on either example falsifies the extension.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper first defines squeezed patterns by direct analogy with squeezed knots, proves every braided pattern is squeezed, then states a τ formula for all squeezed patterns and uses it to obtain both the satellite formulas and the non-homomorphism result for |w|≥2. The argument therefore requires that the key Heegaard Floer or concordance properties (vanishing of certain correction terms or filtration levels) that make the squeezed-knot formula work transfer verbatim when the pattern is placed inside a solid torus and the satellite is formed. No independent derivation or direct HF computation for a non-trivial braided pattern is supplied to confirm the transfer; the homomorphism claim collapses if the formula is off by even one integer on any example.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1811,"tokens_out":357,"duration_ms":21896,"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":[{"comment":"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.","section":null}],"minor_comments":[{"comment":"The introduction could state the explicit τ and ε formulas for braided satellites at the outset rather than deferring them entirely to later sections.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[{"response":"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_made":"partial","referee_comment":"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."}],"tokens_in":1189,"tokens_out":302,"duration_ms":12292,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper gives explicit formulas for the tau and epsilon invariants of satellite knots with braided patterns. It defines squeezed patterns by analogy with squeezed knots, proves every braided pattern is squeezed, states a tau formula for all squeezed patterns, and concludes that no such pattern with winding number at least 2 can be a homomorphism on the concordance group.\n\nThe new pieces are the squeezed-pattern definition in the satellite setting, the proof that braided patterns belong to it, the resulting formulas, and the non-homomorphism claim as progress toward Hedden's conjecture. Explicit, computable values for an infinite family are useful for anyone testing structural questions in concordance.\n\nThe soft spot is the carry-over of the tau formula. The argument treats the key Heegaard Floer properties as transferring directly when the pattern is placed in a solid torus, but the paper supplies no direct computation for a non-trivial braided pattern and no independent check against a known satellite tau value. If the filtration levels or correction terms shift even slightly under the satellite construction, both the formulas and the homomorphism result are off. That step is load-bearing and currently rests on the analogy alone.\n\nThis is for knot theorists already working with Heegaard Floer invariants and satellite constructions. A reader who needs concrete tau values or wants to see concrete movement on homomorphism questions will find material here. The paper engages the literature honestly and makes sharp claims, so it deserves a full referee report rather than a desk rejection.\n\nI would send it to peer review.","headline":"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.","tokens_in":2332,"tokens_out":393,"would_cite":false,"duration_ms":23789,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Braided satellite knots have explicit tau and epsilon formulas, and none with winding number at least two induce concordance homomorphisms.","keywords":["tau-invariant","braided satellites","squeezed patterns","knot concordance","satellite knots","epsilon invariant","winding number","Heegaard Floer homology"],"falsifier":"An explicit braided pattern with winding number two whose induced map on concordance classes is a non-zero homomorphism would refute the central claim.","tokens_in":2517,"feed_emoji":"","tokens_out":558,"duration_ms":21866,"temperature":0.7,"pith_summary":"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.","feed_headline":"Formulas for tau of braided satellites; none induce homs for |w|>=2","feed_subtitle":"The formulas compute invariants for many satellites while the homomorphism result limits algebraic structure in the concordance group.","key_machinery":"Squeezed patterns, the class of solid-torus patterns that admit a tau formula by the same mechanism used for squeezed knots.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Squeezed patterns yield satellite tau; no homs from braided |w|>=2","Formula for tau of squeezed braided satellites; none hom |w|>=2","Braided patterns squeezed; tau expression and homs blocked |w|>=2","Tau of satellites from squeezed; braided patterns don't hom for |w|>=2"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The tau formula known for squeezed knots extends without change to the definition of squeezed patterns inside the solid torus.","fun_headline_variants_meta":{"raw":{"variants":["Squeezed patterns yield satellite tau; no homs from braided |w|>=2","Formula for tau of squeezed braided satellites; none hom |w|>=2","Braided patterns squeezed; tau expression and homs blocked |w|>=2","Tau of satellites from squeezed; braided patterns don't hom for |w|>=2"]},"model":"grok-4.3","cost_usd":0.010536,"raw_usage":{"total_tokens":4588,"prompt_tokens":532,"num_sources_used":0,"completion_tokens":87,"cost_in_usd_ticks":105362000,"prompt_tokens_details":{"text_tokens":532,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3969,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":532,"tokens_out":87,"duration_ms":32762,"temperature":1.0,"reasoning_tokens":3969,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T11:23:06.614689+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit braided pattern with winding number two whose induced map on concordance classes is a non-zero homomorphism would refute the central claim.","supporting_citations":[],"review_version":1}