{"id":"c136f131-6d9f-4ebb-a0fa-4dd3f3f0f996","arxiv_id":"2508.21776","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors construct an n-colored knot Floer homology as a colimit over cable links with increasing full twists and equip it with a module structure over an explicit algebra.","lead":"This paper defines an n-colored version of knot Floer homology for knots, built from limits of cable link Floer homologies. It computes the invariant for unknots and L-space knots and shows it carries an explicit algebraic structure.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central construction depends on external preprint [1] for full-twist cobordism maps: degree formulas and injectivity are not proved in this paper.","rationale":"The paper defines n-colored knot Floer homology as a colimit of link Floer homology of (n,mn)-cables under full-twist cobordism maps φ0. The core content includes a self-contained stabilization theorem (Theorem 1.2), a description of HFL(T(n,mn)) via [5] and [13], a proof that the φ_k satisfy the cable algebra relations (Theorems 4.6 and 5.1), and the localization step identifying H_n(O) with A_n^col (Theorem 4.12). Steps identifying the colimit with stabilized homology, and all module-structure results, rely on Proposition 2.7's degree formulas and injectivity of φ_k, imported from the same-authors preprint [1]. The reader flagged exactly this as the weakest assumption; I agree. If the degree shifts in Proposition 2.7 are wrong, the normalization (4) is wrong and the entire graded structure of H_n(K) collapses—Theorem 1.2 only controls dimensions, not the maps φ0. If injectivity fails, the colimit may be strictly smaller than HFL_stab for large m, invalidating the unknot and L-space computations. The paper's self-contained parts are clearly argued and the examples are consistent, so the central claim is plausible but contingent on external results. A conditional verdict is appropriate rather than rejection or unconditional acceptance.","tokens_in":34613,"tokens_out":13139,"duration_ms":141590,"concrete_test":"Independently derive the grading shifts of φ_k from Zemke's cobordism grading formula ([39, Theorem 1.4]) for the (−1)-framed 2-handle attachment, and compare with Proposition 2.7. Then check injectivity of φ_0 on the explicit Heegaard diagrams of Section 3.1 for n=2, m=1 (i.e., the map HFL(T(2,0))→HFL(T(2,2))) using the generator description in Theorem 4.1. If the derived shifts or the injectivity differ, the normalized grading (4) and the colimit definition (6) are invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing assumption is Proposition 2.7, cited to the same-authors preprint [1]: the cobordism maps φ_k induced by blowing down a (−1)-framed unknot have grw(φ_k)=−k²−k, A_i(φ_k)=−k+(n−1)/2, and are injective on HFL. These facts are not reproduced here. They are used (i) to define the normalized Alexander grading in (4), so that H_n(K) becomes a graded vector space (Theorem 1.1); (ii) in Lemma 4.3 and Corollary 4.4 to identify the unknot colimit H_n(O) with the stabilized homology, and hence Theorem 4.12; (iii) in Theorem 4.6 to prove the φ_k satisfy the cable algebra relations, which underlies Theorem 5.1 and the A_n^col-module structure on H_n(K) for all knots; (iv) in Lemma 6.1 for L-space knots. If the degree formula is off, the normalization (4) fails and the entire graded structure collapses. If injectivity fails, the colimit may be strictly smaller than the stabilized homology, invalidating the unknot and L-space computations. Theorem 1.2 gives a self-contained stabilization of the underlying vector spaces, but it does not identify the connecting maps φ_0, so it does not substitute for [1].","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines a colored knot Floer homology H_n(K) as the colimit, over m, of the link Floer homology of (n,mn)-cables of a knot K, with connecting maps induced by full-twist cobordisms. The main structural results are: (i) a stabilization theorem for the underlying homology groups (Theorem 3.2), giving finite-dimensionality in each grading; (ii) an action of a localized cable algebra A_n^col on the colimit (Theorems 5.1 and 5.2); (iii) a computation of the unknot invariant as a free rank-one module over A_n^col (Theorem 4.12); (iv) a description for L-space knots in terms of HFL(K) tensored with A_n^col (Theorems 6.2 and 6.6); and (v) maps for colored homology of crossing changes (Theorem 7.3). The stabilization argument in Section 3 is self-contained and uses explicit chain-level bijections, but several load-bearing facts about the full-twist cobordism maps are imported from the same-authors preprint [1], and some algebraic steps in the module-structure proofs are incomplete.","tokens_in":34996,"tokens_out":6423,"duration_ms":77862,"significance":"If the cited cobordism-map properties are supplied and the algebraic gaps are repaired, this paper introduces a genuinely new invariant with a rich algebraic structure, connecting link Floer homology to cable algebras and to colored Khovanov-Rozansky homology. The explicit stabilization theorem (Theorem 3.2) is a concrete and useful technical contribution, as are the L-space knot computations and the crossing-change maps. The paper is ambitious and opens several directions, including relations to bordered Floer homology and to colored Khovanov-Rozansky homology. However, the dependence on an unreviewed same-author preprint for the fundamental degree and injectivity formulas is a substantial correctness risk, and the proof of the module action over the cable algebra contains a nontrivial logical gap.","major_comments":[{"comment":"Proposition 2.7 is cited to [1] without proof, but it is load-bearing throughout: the Alexander-degree shift A_i(phi_k) = -k + (n-1)/2 is used in the normalization (4); injectivity of phi_0 is used in Lemma 4.3, Corollary 4.4, Theorem 4.6 Step 1, and Lemma 6.1; and the same facts support Theorems 5.1 and 5.2. Since [1] is a same-authors preprint, this is not an independent check. The authors should either prove these formulas (or at least the special cases used) in the present paper, or explicitly state them as assumptions and mark all downstream results as conditional.","section":"§2.3, Proposition 2.7"},{"comment":"The argument that the cobordism maps phi_k satisfy the linear and quadratic relations (22)-(23) is incomplete. The text says that because both sides have the same Alexander and Maslov degrees they coincide, and that homogeneity of the relations implies the maps satisfy them. But homogeneity alone is not a proof of a relation; and for maps between F[U]-towers, equality of degree shifts determines a nonzero map only up to a unit, so one must also prove nonvanishing or otherwise identify the maps. The injectivity quoted from [1] does not by itself imply the relevant compositions U_I phi_{k-1} and V_I phi_k are nonzero. This step underpins Theorem 5.1 and hence the A_n^col-module structure on H_n(K) (Theorem 1.13), so it needs a complete proof.","section":"§4.2, Theorem 4.6 Step 1"},{"comment":"The explicit presentation of H_n(K) for L-space knots asserts that \"by a similar argument as Theorem 4.1, the relations (28) generate all relations among the generators.\" No proof of completeness is given. Since Theorem 6.2 is the basis for Theorem 1.17 and Theorem 6.6, this gap should be filled, or the statement should be reduced to a conjecture or conditional result.","section":"§6, Theorem 6.2"}],"minor_comments":[{"comment":"The notation is confusing: the same symbol s is used for both the unnormalized and normalized Alexander grading. Please use distinct notation, e.g., s and \\bar{s}, consistently throughout.","section":"§1.2, Eq. (4)"},{"comment":"eA is written as (eA_1,...,eA_m); it should be (eA_1,...,eA_n). Also, the notation n(\\phi) is used without definition; please define it as the total U-exponent.","section":"§3.2, proof of Theorem 3.2"},{"comment":"The definition of A_n^col as the span of Y/a_0^m with tw(Y)=m is not obviously a ring; the subsequent localization statement (Theorem 1.20) fixes this, but it would help to state this explicitly at the definition.","section":"§4, Definition 4.10"},{"comment":"The proof relies on an isotopy changing the order of 2-handle attachment and band attachments, citing [1, Proposition 3.13]. A figure or a more detailed explanation of this isotopy would improve readability and verifiability.","section":"§5.1, Theorem 5.1"},{"comment":"Reference [1] is a preprint by the same authors and is used for several essential facts. The paper should clearly indicate which of its results are proved in [1] and whether [1] is available in a stable, updated form.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's central construction and module structure depend on Proposition 2.7, cited to the same-authors preprint [1]. This is not by itself disqualifying, but editors should ensure that [1] is publicly available and that its proofs are not hidden behind unpublished material. The authors should be asked to either include proofs of the degree formulas and injectivity for the full-twist maps, or clearly state these as assumptions. The gap in Theorem 4.6 Step 1 is also significant and should be fixed before publication. The paper fits the scope of math.GT and would be of interest to the knot Floer homology community."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe paper defines, for each n, an n-colored knot Floer homology H_n(K) as a colimit of link Floer homologies of (n,mn)-cables under full-twist cobordism maps. That construction is genuinely new in the full link Floer setting, and it comes with a rich package: a module structure over the localized cable algebra A_n^col, crossing-change maps, and explicit computations for the unknot and L-space knots. The algebra itself, and its identification with a quotient of a polynomial ring (Theorem 4.12), are clean and convincing. The comparison with colored Khovanov-Rozansky homology is speculative but reasonable.\n\nThe paper is well written and honest about what is proved and what is assumed. The stabilization theorem (3.2) is a self-contained, explicit chain-level argument, and it is the right backbone for finite-dimensionality. The authors do not oversell: they state Conjecture 1.3, and they clearly separate it from the proven stabilization.\n\nThe soft spot is exactly the one flagged in the stress test: Proposition 2.7, which supplies the Alexander and Maslov degree shifts of the cobordism maps and their injectivity, is imported wholesale from [1], a preprint by the same three authors. These facts are load-bearing. They fix the normalization of the Alexander grading, they identify the unknot colimit with the stabilized homology, and they underwrite the algebra relations. Theorem 1.2 stabilizes the vector spaces but says nothing about the connecting maps, so it does not substitute. If the degree formula in [1] is off by a constant, the graded structure collapses; if injectivity fails, the colimit could be smaller than the stabilized homology. That is a real conditional, not a manufactured one. The good news is that the rest of the structure is largely algebraic once you grant those inputs, and the authors are transparent about the dependency.\n\nTwo smaller soft spots: the proof of Theorem 4.1 leans on [5] for the classification of L-space link homology, and the proof that relations (28) generate all relations in Theorem 6.2 is sketched rather than shown. Neither looks fatal, but they add to the sense that parts of the paper are a bridge from existing results rather than fully self-contained.\n\nWho is this for? Anyone working in Heegaard Floer homology, especially on cables and satellite operations, and people interested in categorified knot invariants with algebraic structure. It deserves a serious referee. If I were the editor, I would send it out, with referee instructions to check the dependence on [1] and, ideally, a request that the authors either include the needed statements from [1] or clearly state which version they rely on.","headline":"A genuinely new invariant with a rich algebraic structure; the main caveat is that its well-definedness rests on a same-authors preprint, so it deserves serious refereeing, not desk rejection.","tokens_in":35462,"tokens_out":2535,"would_cite":true,"duration_ms":28517,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K18","57K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Infinite full-twist limits of cable Floer homology define a triply graded knot invariant that is a module over an explicit algebra, computed exactly for unknots and L-space knots.","keywords":["colored knot Floer homology","link Floer homology","cable links","full twists","cable algebra","L-space knots","colimits","Khovanov-Rozansky homology"],"falsifier":"Compute the link Floer homology of the $(2,2m)$ cables of the figure-eight knot for $m = 1$ through $6$ by bordered Floer homology, apply the renormalization $s = s - (c_m,c_m)$, and test the stabilization isomorphisms of Theorem 1.2 degree by degree; then check whether the stabilized groups carry the $A^{\\mathrm{col}}_2$ action with $A$ of Maslov degree $-2$ and Alexander degree $(-1,-1)$. Failure in any fixed Alexander degree would disprove Theorem 1.2 or Theorem 1.13 for non-L-space knots. A milder check is an independent verification of Proposition 2.7's degree formula on the torus link change $T(2,2) \\to T(2,4)$, using","tokens_in":34549,"feed_emoji":"🧶","tokens_out":18430,"duration_ms":178453,"temperature":0.7,"texified_at":"2026-08-05T20:18:34.565621+00:00","pith_summary":"Colored invariants label a knot by a representation — here, the $n$-dimensional one — and in quantum topology one realizes such a colored invariant as a limit of ordinary invariants of cables. This paper performs that move inside knot Floer homology: it takes the $(n, mn)$-cable of a knot $K$ ($n$ parallel copies, pairwise linking number $m$), computes the full link Floer homology, and passes to the colimit as $m$ tends to infinity along maps that insert one additional full twist. The paper proves this infinite-full-twist limit is a well-defined triply graded invariant, finite-dimensional in every degree, and a module over an explicitly presented algebra $A^{\\mathrm{col}}_n$ with generators $U_1,\\ldots,U_n, V_1,\\ldots,V_n, A$ and relations $U_i = A \\prod_{j \\neq i} V_j$. The invariant is computed exactly for the unknot (it is $A^{\\mathrm{col}}_n$ itself) and for L-space knots (it is $H_{FL}(K)$ tensored with $A^{\\mathrm{col}}_n$), and maps for crossing changes are constructed. If the construction is sound, knot Floer homology gains a colored companion parallel to the $S^n$-colored Khovanov–Rozansky homology, organizing the full-twist behavior of all cable homologies into a single module.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":8404,"prompt_tokens":1094,"completion_tokens":7310,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":1094,"completion_tokens_details":{"reasoning_tokens":6157}},"feed_headline":"Infinite full twists define colored knot Floer homology","feed_subtitle":"The invariant is a module over an explicit algebra, with exact answers for unknots and L-space knots.","key_machinery":"The $n$-strand cable algebra $A_n$: a triply graded algebra over $F[U_1,\\ldots,U_n,V_1,\\ldots,V_n]/(U_i V_i = U_j V_j)$ with commuting generators $a_0,\\ldots,a_{n-1}$, subject to linear relations $U_I a_{k-1} = V_{\\bar I} a_k$ $(|I| = k)$ and quadratic relations $a_i a_j = U^{k\\ell - i j} a_k a_\\ell$ $(i+j = k+\\ell, i \\leq k \\leq \\ell \\leq j)$. The generators $a_k$ are realized geometrically as the full-twist cobordism maps $\\phi_k$; the localization $A^{\\mathrm{col}}_n = A_n[a_0^{-1}] \\cong F[U_1,\\ldots,U_n,V_1,\\ldots,V_n,A]/(U_i = A \\prod_{j \\neq i} V_j)$, with $A = a_1/a_0$, is the algebra acting on colored homology, where $A$ shifts Alexander degree by $(-1,\\ldots,-1)$ and Maslov degree by $-2$. Carrying the argument is the Alexander-degree normalization $s = s - (c_m,\\ldots,c_m)$, $c_m = m(n$","core_discovery":"The central claim is that the colimit $H_n(K) = \\mathrm{colim}_m H_{FL}(K_{n,mn})$ along full-twist cobordism maps $\\phi_0$ is a well-defined triply graded invariant with a module structure. With $c_m = m(n-1)/2$ the maps $\\phi_0$ preserve the shifted Alexander grading $s = s - (c_m,\\ldots,c_m)$, and for fixed $s$ the stabilized groups $H_{FL}^{\\mathrm{stab}}(K_{n,mn}, s)$ are isomorphic for all large $m$, proved by an explicit bijection between generators of special Heegaard diagrams. The full-twist maps $\\phi_k$, obtained by blowing down a $(-1)$-framed unknot in different $\\mathrm{Spin}^c$ structures, satisfy the relations of the cable algebra $A_n$, whose localization $A^{\\mathrm{col}}_n = A_n[a_0^{-1}]$ acts on $H_n(K)$. Two cases are computed: the unknot, $H_n(O) \\cong A^c$","pith_inferences":["The tensor-product formula for L-space knots reads like a base change of ordinary knot Floer homology from F[U,V] to A^col_n along U↦A, V↦V_1...V_n. A natural question the paper does not pose: is H_n characterized as the universal A^col_n-module extending H_FL(K) compatibly with cobordism maps — a property that, if true, would make the colored theory a change of base ring rather than genuinely new","The stabilization proof in Theorem 3.2 is a combinatorial statement about special Heegaard diagrams. If the same diagrammatic argument extends to the (n, mn+r) families of Conjecture 1.11, the colimit construction would yield colored invariants for every cable slope and would connect the full-version theory here to the hat-version construction of [9].","Lemma 1.24 shows the A^col_n relations are exactly a specialization of the relations of the 'y-ified' colored HOMFLY-PT homology of [2, 12]. The concrete test this suggests: compare the graded Euler characteristics of H_FL(T(n,mn)) from Theorem 4.1 with the specialized Poincaré polynomials of the y-ified unknot theory for small n and m, to see whether both stabilizations give the same limit.","Read through the sheaf interpretation of Section 1.6, Problem 1.14 — finite generation of H_n(K) over A^col_n — is a finite-type statement for the quasi-coherent sheaf on the chart {a_0 ≠ 0}; the L-space knot theorem is the first family where it holds. A testable route to the general case would be to prove the connecting maps φ_0 are eventually injective in each renormalized degree, using the stab"],"forward_implications":["Colored knot Floer homology is a genuine knot invariant: for every knot K and every n ≥ 1, H_n(K) is a triply graded F-vector space with finite-dimensional pieces in every Zn⊕Z-degree.","The unknot computation fixes the target ring: since H_n(O) ≅ A^col_n, every colored invariant is a module over an algebra with an explicit presentation, so questions about colored invariants become questions about modules over a known ring.","For L-space knots the colored invariant is determined by ordinary knot Floer homology: H_n(K) ≅ H_FL(K) ⊗_{F[U,V]} A^col_n with explicit generators and relations, giving a large family of fully computable examples.","Crossing changes act on colored homology: there are maps G^col_j: H_n(K_-) → H_n(K_+) and F^col_j: H_n(K_+) → H_n(K_-) of Maslov degree -j²-j that commute with the A^col_n action, a colored analogue of the usual skein-type cobordism maps.","Assuming Conjecture 1.3 (the connecting maps are eventually isomorphisms), the Euler characteristic of H_n(K) is (t_1...t_n)^{1/2}χ_K(t_1...t_n), so the colored limit recovers the Alexander polynomial of K, the expected categorified Rosso–Jones relation."],"supporting_citations":[{"why":"Supplies Proposition 2.7: the degree shifts (Maslov -k²-k, Alexander -k+(n-1)/2 per component) and injectivity of the full-twist cobordism maps φ_k, plus the crossing-change cobordism maps used in Section 7.","marker":"[1]"},{"why":"Zemke's functoriality theorem for decorated link cobordisms, which provides the cobordism maps φ_k and the naturality used to transfer the cable-algebra relations from the unlink to arbitrary links in Theorem 5.1.","marker":"[38]"},{"why":"Zemke's absolute grading change formulas for link cobordism maps, used to compute the degrees in Proposition 2.7 and the Alexander-degree shifts in Lemma 3.4.","marker":"[39]"},{"why":"Borodzik–Liu–Zemke's lattice-homology description of H_FL for plumbed L-space links, the basis for the explicit computation of H_FL(T(n,mn)) in Theorem 4.1 and its F[U]-towers.","marker":"[5]"},{"why":"Gorsky–Hom's criterion that large cables of L-space knots are L-space links and the h-function formula, used to compute H_FL(K_{n,mn}) and to prove the L-space knot description in Lemma 6.1 and Theorem 6.6.","marker":"[13]"},{"why":"Hedden's cabling results for the (n, mn+1) families, the precedent for Conjecture 1.11(a) and the source of the Heegaard diagrams the authors adapt to (n, mn) cables in Section 3.","marker":"[16]"},{"why":"Rozansky's infinite-torus-braid construction, the Khovanov–Rozansky 'infinite full twist' colimit that motivates Definition 1.6 of colored knot Floer homology.","marker":"[35]"}],"fun_headline_variants":["Colored knot Floer homology defined by infinite full twists","Infinite full twists give module structure on knot Floer homology","Colored knot Floer homology: exact answers for L-space knots","New invariant from cable limits: colored knot Floer homology","Colimit of cable Floer homology yields colored knot invariant"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The construction depends on the authors' companion preprint for the claim that each full-twist cobordism map $\\phi_k$ shifts Maslov degree by $-k^2 - k$ and each Alexander degree by $-k + (n-1)/2$ and is injective; if those grading shifts are off, the renormalized colimit and the whole module structure fail.","fun_headline_variants_meta":{"raw":{"variants":["Colored knot Floer homology defined by infinite full twists","Infinite full twists give module structure on knot Floer homology","Colored knot Floer homology: exact answers for L-space knots","New invariant from cable limits: colored knot Floer homology","Colimit of cable Floer homology yields colored knot invariant"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000213,"raw_usage":{"total_tokens":1249,"prompt_tokens":729,"completion_tokens":520,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":473,"completion_tokens_details":{"reasoning_tokens":445}},"tokens_in":473,"tokens_out":520,"duration_ms":6395,"temperature":1.0,"reasoning_tokens":445,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T13:56:34.582366+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the link Floer homology of the $(2,2m)$ cables of the figure-eight knot for $m = 1$ through $6$ by bordered Floer homology, apply the renormalization $s = s - (c_m,c_m)$, and test the stabilization isomorphisms of Theorem 1.2 degree by degree; then check whether the stabilized groups carry the $A^{\\mathrm{col}}_2$ action with $A$ of Maslov degree $-2$ and Alexander degree $(-1,-1)$. Failure in any fixed Alexander degree would disprove Theorem 1.2 or Theorem 1.13 for non-L-space knots. A milder check is an independent verification of Proposition 2.7's degree formula on the torus link change $T(2,2) \\to T(2,4)$, using","supporting_citations":[{"cited_title":"Splitting maps in link Floer homology and integer points in permutahedra","cited_arxiv_id":"2307.07741","evidence_quote":"Supplies Proposition 2.7: the degree shifts (Maslov -k²-k, Alexander -k+(n-1)/2 per component) and injectivity of the full-twist cobordism maps φ_k, plus the crossing-change cobordism maps used in Section 7."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Zemke's functoriality theorem for decorated link cobordisms, which provides the cobordism maps φ_k and the naturality used to transfer the cable-algebra relations from the unlink to arbitrary links in Theorem 5.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Zemke's absolute grading change formulas for link cobordism maps, used to compute the degrees in Proposition 2.7 and the Alexander-degree shifts in Lemma 3.4."},{"cited_title":"Lattice homology, formality, and plumbed L-space links","cited_arxiv_id":"2210.15792","evidence_quote":"Borodzik–Liu–Zemke's lattice-homology description of H_FL for plumbed L-space links, the basis for the explicit computation of H_FL(T(n,mn)) in Theorem 4.1 and its F[U]-towers."},{"cited_title":"Gorsky, J","cited_arxiv_id":null,"evidence_quote":"Gorsky–Hom's criterion that large cables of L-space knots are L-space links and the h-function formula, used to compute H_FL(K_{n,mn}) and to prove the L-space knot description in Lemma 6.1 and Theorem 6.6."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Hedden's cabling results for the (n, mn+1) families, the precedent for Conjecture 1.11(a) and the source of the Heegaard diagrams the authors adapt to (n, mn) cables in Section 3."},{"cited_title":"Rozansky","cited_arxiv_id":null,"evidence_quote":"Rozansky's infinite-torus-braid construction, the Khovanov–Rozansky 'infinite full twist' colimit that motivates Definition 1.6 of colored knot Floer homology."}],"review_version":1}