{"id":"e810db73-e5d2-4523-8392-fd190e4f9a78","arxiv_id":"2606.30823","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Authors produce a class of deeply slice knots in non-trivial integral homology 3-spheres by building contractible 4-manifolds where the knots are slice and verifying they fail to be slice in the 3-manifold times interval via immersed-curve Heegaard Floer invariants.","lead":"The paper constructs examples of knots that are slice in a contractible 4-manifold bounded by a homology 3-sphere but not slice in the product of that 3-sphere with an interval. This makes concrete progress on an open question from Akbulut on the Kirby list by using Heegaard Floer concordance invariants computed via immersed curves.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"Immersed-curve extraction of HF concordance invariants may fail to certify non-sliceness if the curve-to-invariant map contains an unstated assumption for these examples","rationale":"The reader's weakest assumption is precisely the load-bearing computational step. No other part of the argument (existence of the contractible filling, homology-sphere boundary, or 0-homology class) appears internally inconsistent on the basis of the abstract and described method. The concern is therefore isolated to verification of the invariant extraction, which the proposed test directly addresses.","tokens_in":1676,"tokens_out":329,"duration_ms":10768,"concrete_test":"Take the first explicit knot and its immersed curve from the paper; recompute the relevant HF concordance invariant by an independent method (e.g., via the standard Heegaard diagram or software) and check whether the numerical value matches the immersed-curve extraction; mismatch falsifies the obstruction claim for that example.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the constructed knots have non-vanishing (or otherwise obstructing) HF concordance invariants in Y×[0,1]. The paper obtains these invariants exclusively via immersed-curve techniques. If the step that converts the immersed curve data into the numerical invariant (e.g., the value used to obstruct sliceness) relies on an implicit normalization, grading convention, or identification that does not hold for the specific curves arising from the contractible fillings, then the obstruction is not established. This is the only place where the argument moves from geometric construction to certified non-sliceness.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper constructs a class of knots K in integral homology 3-spheres Y ≠ S³ such that K is slice in a contractible 4-manifold X with ∂X = Y but is not slice in Y × [0,1]. The knots are obtained by specifying pairs (X, K) with K slice in X; non-sliceness in the product is certified by Heegaard Floer concordance invariants extracted from immersed-curve diagrams of the knot complements.","tokens_in":1821,"tokens_out":383,"duration_ms":19533,"significance":"If the immersed-curve computations are verified to produce obstructing invariants, the examples furnish the first explicit negative answer to Akbulut’s question on the Kirby list and demonstrate that HF concordance invariants can distinguish sliceness in different fillings of the same 3-manifold. The use of immersed curves for explicit computation is a methodological strength that could be reusable for other concordance questions.","major_comments":[{"comment":"The central claim requires that the extracted HF concordance invariants are non-vanishing (or otherwise obstructing) for the constructed knots. The manuscript must therefore supply, for at least one explicit example, the immersed-curve diagram, the precise map from curve data to the numerical invariant used for the obstruction, and a verification that the grading/identification conventions employed are valid for the curves arising from the contractible filling. Without these explicit data the obstruction step remains unverified.","section":"Section describing invariant extraction from immersed curves"}],"minor_comments":[{"comment":"The abstract states the existence of a class but supplies no concrete manifold, knot diagram, or invariant value; moving at least one explicit example into the abstract or introduction would improve readability.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their thorough review and for identifying the need for greater explicitness in the invariant computations. We address the single major comment below and will incorporate the requested material into a revised manuscript.","responses":[{"response":"We agree that an explicit worked example is necessary to make the obstruction step fully verifiable. In the revised manuscript we will add, in the section on invariant extraction, at least one complete example consisting of: (i) the immersed-curve diagram for the knot complement arising from the contractible filling, (ii) the precise algebraic map that converts the curve data into the numerical concordance invariant, and (iii) a short verification that the grading and identification conventions match those required by the contractible 4-manifold. These additions will be placed immediately after the general description of the invariant so that readers can check the non-vanishing claim directly.","revision_made":"yes","referee_comment":"[Section describing invariant extraction from immersed curves] The central claim requires that the extracted HF concordance invariants are non-vanishing (or otherwise obstructing) for the constructed knots. The manuscript must therefore supply, for at least one explicit example, the immersed-curve diagram, the precise map from curve data to the numerical invariant used for the obstruction, and a verification that the grading/identification conventions employed are valid for the curves arising from the contractible filling. Without these explicit data the obstruction step remains unverified."}],"tokens_in":1284,"tokens_out":313,"duration_ms":19490,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper targets Akbulut's question on the Kirby list: whether every knot in a homology sphere Y that is slice in some contractible filling X must already be slice in the product Y x I. They claim to produce a class of deeply slice knots that are slice in X but not in the product, detected by Heegaard Floer concordance invariants extracted from immersed curves.\n\nWhat is new is the specific class of examples in non-trivial Y. The immersed-curve technique itself comes from prior work, but directing it at this named question and claiming explicit progress counts as the advance. If the calculations are correct, it supplies concrete instances rather than another existence argument.\n\nThe paper frames the problem cleanly and states the method without overclaiming. That is the main positive.\n\nThe soft spot is verification. The abstract describes the construction and the use of immersed curves but gives no manifolds, no diagrams, and no sample invariant values showing the obstruction. The stress-test note is on point here: the move from curve data to the numerical invariant that rules out sliceness in the product could rest on an unstated grading or normalization that fails for these particular curves. Without the body, that step cannot be checked.\n\nThis is for people already working in 4-manifold topology and Heegaard Floer concordance who follow Akbulut-type questions or immersed-curve methods. A reader outside that circle will not get much. It is worth sending to a serious referee so the constructions and the invariant extraction can be examined directly.","headline":"They construct a class of deeply slice knots in non-S3 homology spheres using immersed curves for HF concordance invariants, but the abstract supplies no explicit examples or computations to check the obstruction.","tokens_in":2285,"tokens_out":391,"would_cite":false,"duration_ms":26732,"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":"Knots in non-trivial homology spheres can be slice in a contractible 4-manifold but not in the product cobordism, as certified by immersed-curve Heegaard Floer invariants.","keywords":["deeply slice knots","Heegaard Floer homology","immersed curves","concordance invariants","contractible 4-manifolds","homology 3-spheres","Akbulut question"],"falsifier":"An explicit example where the immersed-curve invariant vanishes for one of the constructed knots yet an independent method shows the knot is slice in Y × [0,1], or a direct computation proving the knot is not slice in any contractible filling.","tokens_in":2585,"feed_emoji":"","tokens_out":696,"duration_ms":26796,"temperature":0.7,"pith_summary":"The paper constructs a class of knots in integral homology 3-spheres Y other than S^3 that are slice in some contractible 4-manifold X bounded by Y. These knots are shown not to be slice in the product Y times an interval. The authors address a question posed by Akbulut by using concordance invariants from Heegaard Floer homology, computed through immersed curve techniques, to establish the distinction. This produces examples of deeply slice knots and makes progress on whether sliceness in a contractible filling implies sliceness in the product.","feed_headline":"Immersed curves detect knots slice only in contractible 4-manifolds","feed_subtitle":"Examples in homology spheres other than S^3 are not slice in the product Y x I, certified by Heegaard Floer invariants.","key_machinery":"Immersed curve techniques applied to Heegaard Floer homology to extract concordance invariants that obstruct sliceness in the product cobordism.","core_discovery":"There exists a class of knots K in integral homology 3-spheres Y ≠ S³ such that K is slice in some contractible 4-manifold X bounded by Y but is not slice in Y × [0,1], detected by Heegaard Floer concordance invariants computed via immersed curves.","pith_inferences":["These examples suggest that the distinction between different 4-dimensional fillings may be detectable in higher invariants for a wider range of 3-manifolds.","The technique could be tested on other contractible manifolds with the same boundary to see if the obstruction persists.","If the invariants can be made algorithmic for larger families, they might classify which knots admit such asymmetric sliceness."],"forward_implications":["Such knots separate the notion of sliceness in a contractible filling from sliceness in the product cobordism for homology spheres.","The construction supplies negative instances for Akbulut's question on the Kirby list.","Immersed curve methods become a practical tool for certifying non-sliceness in concordance questions beyond the 3-sphere.","Further examples can be generated by varying the pair (X, K) while preserving the boundary homology sphere."],"fun_headline_variants":["Immersed curves detect deeply slice knots in homology spheres","Heegaard Floer via immersed curves flags knots not slice in Y times I","Knots slice in contractible manifolds but not homology sphere products","Immersed curves reveal class of deeply slice knots in non S3 spheres"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The immersed-curve computations of the Heegaard Floer concordance invariants correctly certify that the constructed knots fail to be slice in Y × [0,1].","fun_headline_variants_meta":{"raw":{"variants":["Immersed curves detect deeply slice knots in homology spheres","Heegaard Floer via immersed curves flags knots not slice in Y times I","Knots slice in contractible manifolds but not homology sphere products","Immersed curves reveal class of deeply slice knots in non S3 spheres"]},"model":"grok-4.3","cost_usd":0.008207,"raw_usage":{"total_tokens":3687,"prompt_tokens":593,"num_sources_used":0,"completion_tokens":76,"cost_in_usd_ticks":82074500,"prompt_tokens_details":{"text_tokens":593,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3018,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":593,"tokens_out":76,"duration_ms":31746,"temperature":1.0,"reasoning_tokens":3018,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-01T01:26:08.234366+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit example where the immersed-curve invariant vanishes for one of the constructed knots yet an independent method shows the knot is slice in Y × [0,1], or a direct computation proving the knot is not slice in any contractible filling.","supporting_citations":[],"review_version":1}