{"id":"f7fe2a4e-9dff-4aa0-9206-c90cd283b2d0","arxiv_id":"2506.09296","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Octahedral fully augmented link complements are isometric to fundamental shadow link complements, and the Turaev-Viro volume conjecture for these links is re-proved via a skein-theoretic coloured Jones formula.","lead":"The paper gives a new geometric proof that octahedral fully augmented link complements are the same hyperbolic spaces as fundamental shadow link complements, and uses it to re-verify the Turaev-Viro volume conjecture for these links. The value is a more geometric and diagrammatic route to a volume conjecture already known to hold for this family.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.15's sign-uniformity proof relies on Eq. (4.17), whose denominator is misprinted for nonzero j1,j2; as printed the formula is undefined, so Corollary 4.20 and the lower-bound step of the TV proof are not yet substantiated.","rationale":"We agree with the reader that the weakest link is Lemma 4.15. The statement of the main theorem (TV volume conjecture for flat octahedral FALs) is already known from [2,38], so correctness is not the issue; the paper's contribution is the new proof, and that proof is gated by the sign-uniformity lemma. The concrete typo in (4.17) is exactly the kind of issue that can be fixed, but as printed it invalidates the proof of Lemma 4.15. We considered the informal diagrammatic arguments in Lemma 2.18; they are a stylistic concern but not a definite mathematical error, whereas (4.17) is a definite error in a central displayed formula. The proposed check (correct the denominator and re-run the sign computation, including numerical small-r spot checks) would settle whether the sign uniformity claim is true. If the corrected computation confirms the lemma, the paper's conditional acceptance is appropriate; if not, the lower bound in Lemma 4.25 fails and the new TV proof would need significant repair. Because the reader already identified this concern and rendered CONDITIONAL, we recommend no change to the verdict.","tokens_in":24589,"tokens_out":12243,"duration_ms":112287,"concrete_test":"Correct (4.17) by replacing [n_r - j1/2 - j2/2 - z]! with [n_r + (j1+j2)/2 - z]!, then recompute the sign of the 6j-symbol {n_r n_r j1; n_r n_r j2} from Definition 4.8 for representative odd r (say r=5,7,9,11) and all admissible j1,j2 in {0,2,...,r-3}, checking whether the sign is +1 for r≡3 mod 4 and -1 for r≡1 mod 4. If any value deviates, Lemma 4.15 and Corollary 4.20 fail. If all match, the typo is confirmed as the only obstruction and the proof can be repaired.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's new TV-volume verification stands on the lower bound in Lemma 4.25, which uses Corollary 4.20 to conclude all summands N_{n_r,...,n_r,j1,...,jc} have the same sign. Corollary 4.20 depends on Lemma 4.15, whose proof computes the sign of the quantum 6j-symbol via Eq. (4.17). In (4.17), the last squared denominator factor is written as [n_r - j1/2 - j2/2 - z]!. For any z in the summation range with j1 or j2 nonzero, this argument is negative, so the quantum factorial is undefined and the displayed expression is not a valid specialization of Definition 4.8. From the general formula (4.11)-(4.13), that factor should be [n_r + (j1+j2)/2 - z]! (from Q2 = Q3 = n_r + (j1+j2)/2). The sign computation ignores squared factors as positive, so the typo may be benign once corrected, but the published proof does not establish the needed sign uniformity. The zero-color cases are handled separately and do not fix the nonzero case. Without Lemma 4.15, the truncation to a single summand in Lemma 4.25 is unjustified, and the claimed geometric proof of the TV volume conjecture for flat octahedral fully augmented links is incomplete.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper has three main threads. First, it gives a new, geometric proof (Theorem 2.12) that for an octahedral fully augmented link L in S^3 with c crossing circles, the complement S^3 - L is isometric to the complement of a fundamental shadow link in #^c(S^1 x S^2). The proof uses the circle-packing description of fully augmented links, translates central subdivision of the nerve into a graph move on the associated 4-valent gluing graph (Lemmas 2.15 and 2.18), and connects the result to the change-of-pair operation of Wong and Yang (Proposition 2.20). Second, the paper derives a formula for the coloured Jones polynomial of octahedral fully augmented links using Kauffman-bracket recoupling theory (Proposition 3.16): the evaluation is a sum over c merging colours of products of Delta-lambda factors, optional half-twist factors, and exactly c-1 quantum 6j-symbols, with all trihedron coefficients cancelling. Third, using the Detcherry-Kalfagianni-Yang formula relating Turaev-Viro invariants to sums of squared coloured Jones polynomials (Theorem 4.1), the paper proves the TV volume conjecture for flat octahedral fully augmented links (Theorem 4.26) and a related asymptotic for the coloured Jones polynomial (Theorem 4.28).","tokens_in":24844,"tokens_out":40233,"duration_ms":342402,"significance":"The correspondence of Theorem 2.12 is proved as an isometry rather than a homeomorphism, which is genuinely useful: it identifies a diagrammatic bridge between the two octahedral families, and the graph-move induction gives an explicit mechanism that the authors reuse for the skein-theoretic computation. The coloured Jones formula of Proposition 3.16 is a clean structural result (one quantum 6j-symbol per octahedron, with all trihedron factors cancelling), and the sign-uniformity method of Lemma 4.15/Corollary 4.20 is an original approach to the lower bound in the TV volume conjecture; Theorem 4.28 answers a question of Detcherry-Kalfagianni-Yang for this family in the even-colouring case. The paper is exemplary in attribution: it states plainly that Theorem 4.26 follows from [2,38] and that the half-twist case remains open (Remark 4.29). The new proofs are only partially delivered, however: the sign computation in Lemma 4.15 rests on a misprinted formula (Eq. (4.17)) that is undefined as written, and the gluing argument in Lemma 2.18 is diagrammatic rather than isometry-level.","major_comments":[{"comment":"Equation (4.17) is not a valid specialization of Definition 4.8, so the proof of Lemma 4.15 does not establish the sign uniformity on which the lower bound in Lemma 4.25 rests. For the 6-tuple (n_r,n_r,j_1,n_r,n_r,j_2), Definition 4.8 gives T_1 = T_4 = n_r + j_1/2, T_2 = T_3 = n_r + j_2/2, Q_1 = 2n_r, and Q_2 = Q_3 = n_r + (j_1+j_2)/2, so the factors [Q_2-z]![Q_3-z]! in (4.11) equal ([n_r+(j_1+j_2)/2-z]!)^2. The displayed (4.17) instead has ([n_r-j_1/2-j_2/2-z]!)^2. For any z in the summation range with j_1 or j_2 nonzero, the argument n_r-j_1/2-j_2/2-z is negative, and the quantum factorial is not defined for negative arguments, so the displayed S^{j_1,j_2}_z is meaningless. Since the sign computation for S^{j_1,j_2}_z feeds directly into Lemma 4.15, then into Corollary 4.20 (sign of N independent of the j_k), and then into the single-summand truncation in Lemma 4.25, the new proof of Theorem 4.26 (and Theorem 4.28) is incomplete as printed. The factor is squared in (4.17), so replacing it by ([n_r+(j_1+j_2)/2-z]!)^2 likely preserves the subsequent parity argument; but as published the computation does not go through and must be corrected and rechecked.","section":"§4, Eq. (4.17), Lemma 4.15"},{"comment":"The proof of Lemma 2.18, which is the induction step for Theorem 2.12, is carried out by inspection of Figures 9 and 10 rather than by an explicit comparison of gluing isometries, and several load-bearing assertions are not justified in the text. (i) The claim that the second ideal vertex created by a central subdivision (the red vertex in Figure 10) must correspond to a crossing circle is made by reference to the figure. (ii) The claim that the gluing of the new octahedron O_1 is forced, namely that the final shaded face of O_1 must be glued to S', uses the unproved assertion that every shaded face is glued to a distinct shaded face. (iii) The assignment of gluing data on the two free edges and the loop of the new gluing graph is asserted to match Definition 2.17, but no argument is given that the edge adjacent to S is glued by the identity and that the loop label is the reflection across the edge joining the two triangular faces. (iv) The case v_2 = v_3, where the deleted edge e is a loop, is noted parenthetically but the graph move is not worked out for a loop. Since the statement of Theorem 2.12 is already known from [38], this is a rigor gap in the paper's new geometric proof rather than an error in a theorem; the authors should supply an isometry-level argument or state explicitly which parts of the correspondence are obtained by comparison with [38].","section":"§2.3, Lemma 2.18"}],"minor_comments":[{"comment":"Please fix the following typos: 'Kaufman multi-bracket' should be 'Kauffman multi-bracket' (Definition 3.1); 'there arec−1 quantum 6j-symbols' is missing a space (proof of Proposition 3.16); 'th even integers' should be 'the even integers' (Theorem 4.28); and 'triangluations' should be 'triangulations' (introduction, first paragraph).","section":"§3.2, Definitions 3.1 and 3.16"},{"comment":"The notation J_{L,iii+111}(A) and '(iii+111)th coloured Jones polynomial' is confusing because the bold multi-index does not survive typesetting; please introduce the multi-index \\mathbf{i} and write (\\mathbf{i}+\\mathbf{1})-coloured explicitly.","section":"§3.1, Definition 3.1"},{"comment":"Lemma 4.25 states 'for odd r', but for r = 3 the colour n_r equals 0, which is outside the range 1 ≤ i_k ≤ m of the summation in Theorem 4.1; the proof should explicitly restrict to r ≥ 5 (or to all sufficiently large odd r), which is harmless for the limit.","section":"§4, Lemma 4.25"},{"comment":"In Lemma 2.15, the sentence 'Two of the Borromean twisted sisters have homeomorphic complements' is asserted without proof or citation; please add a reference or a sketch of the isotopy.","section":"§2.3, Lemma 2.15"},{"comment":"The trihedron-cancellation bookkeeping in the proof of Proposition 3.16 is intricate and difficult to verify from the text; a systematic tracking of when each 3-vertex is created and removed (for example, a table indexed by the triangle pops) would make the argument checkable. This point is not load-bearing for the asymptotic results, since trihedron factors contribute only O(log r/r).","section":"§3.2, Proposition 3.16"},{"comment":"The assertion in the proof of Theorem 2.12 that gluing the c−1 building blocks 'gives a genus c handlebody' is made without argument; a one-line Euler-characteristic computation would clarify the count.","section":"§2.3, proof of Theorem 2.12"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the main theorems are already in the literature ([2,38]), so the paper's value lies entirely in the new proof mechanisms; if those proofs remain informal, the paper is essentially a survey with variations. In my view both issues are fixable within the manuscript's scope: (4.17) is a local misprint whose correction appears to preserve the sign computation, and Lemma 2.18 can be made rigorous with an explicit isometry-level check. I would therefore not reject, but I would require a careful revision rather than minor edits. I did not independently verify the bounds quoted from [2] and [16]; they are standard and the dependence is correctly cited."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a useful, honest paper that does what it says. The headline TV volume conjecture verification for flat octahedral fully augmented links is not new — it was already known from Belletti–Detcherry–Kalfagianni–Yang and Wong–Yang — but the paper gives a genuinely geometric proof of the isometry correspondence and an explicit skein-theoretic formula for the coloured Jones polynomials. Those are real contributions.\n\nThe isometry theorem (Theorem 2.12) upgrades the known homeomorphism to an actual isometry between octahedral FAL complements and fundamental shadow link complements. The circle-packing proof by induction on central subdivision is natural and mostly convincing. Lemma 2.18 is the one place I wanted more detail: the gluing of the new octahedron's shaded faces is described verbally and with diagrams, and it would be easy for a subtle case to hide there. But the argument is plausible and the statement is supported by the later change-of-pair comparison with Wong–Yang, so I don't think it is load-bearing.\n\nProposition 3.16, the coloured Jones formula, is the other solid piece. The cancellation of trihedron coefficients is carefully argued, and the claim that each octahedron contributes exactly one quantum 6j-symbol matches the geometry. I did not check every index, but the structure is coherent.\n\nThe real problem sits in Lemma 4.15. Equation (4.17) contains a denominator factor [n_r - j1/2 - j2/2 - z]!, which is negative for z in the summation range when j1 and j2 are nonzero, so the quantum factorial is undefined. The stress-test note is correct. The intended factor is [n_r + (j1+j2)/2 - z]!, as you get from Q2 and Q3 in Definition 4.8. Since that factor is squared, correcting it does not affect the sign computation, so the lemma is likely repairable with a one-line change. But as published, Corollary 4.20 and the lower bound in Lemma 4.25 rest on an undefined expression. That is a fixable gap, not a dead end, but it needs to be fixed before the proof is accepted.\n\nThe paper is honest about the half-twist obstruction, which is a real limitation. The upper bound works with half-twists, the lower bound does not, and they say so explicitly. That is the right way to handle it.\n\nOverall: the main theorem is already known, so this does not open new land. But the isometry and the Jones formula are useful new tools, and the TV proof is a different route that mostly works modulo a fixable typo. I would send it to a serious referee rather than desk reject. The referee should ask for the Eq. (4.17) correction and a tighter write-up of Lemma 2.18, but should not demand new mathematics.","headline":"Solid incremental paper: the new isometry proof and explicit coloured Jones formula are useful, and the TV volume conjecture proof is a known result whose new route mostly works once a fixable typo in Eq. (4.17) is corrected.","tokens_in":25426,"tokens_out":3730,"would_cite":true,"duration_ms":36748,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K10","57K12","57K16","57K31","57K32"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that every octahedral fully augmented link complement is isometric to a fundamental shadow link complement, and uses this identification to verify the Turaev–Viro volume conjecture for the flat case.","keywords":["fully augmented links","fundamental shadow links","Turaev–Viro volume conjecture","quantum 6j-symbols","coloured Jones polynomial","circle packings","hyperbolic volume","regular ideal octahedra"],"falsifier":"Compute the sign of $S_z^{j_1,j_2}$ in equation (4.17) for a small admissible case such as $r=7$ (so $n_r=2$) with pairs like $(j_1,j_2)=(2,4)$ or $(4,2)$, using the corrected denominator $n_r+(j_1+j_2)/2-z$ in place of the printed $n_r-j_1/2-j_2/2-z$; if the sign depends on $j_1,j_2$ rather than only on $r$, Lemma 4.15 is false. Alternatively, evaluate $|TV_r(S^3\\setminus L,e^{2\\pi i/r})|$ numerically for a small flat octahedral fully augmented link with $c=3$ crossing circles and check whether the limit equals $4v_8$.","tokens_in":24342,"feed_emoji":"🪢","tokens_out":8482,"duration_ms":88522,"temperature":0.7,"pith_summary":"This paper establishes a geometric identification between two families of hyperbolic link complements built from regular ideal octahedra: octahedral fully augmented links in the 3-sphere and fundamental shadow links in connected sums of $S^1\\times S^2$. The authors prove that the complements are isometric, not merely homeomorphic, which gives an explicit volume $2(c-1)v_8$ for a link with $c$ crossing circles. They then compute the coloured Jones polynomial of these links via skein and recoupling theory, obtaining a formula in which each octahedron contributes exactly one quantum $6j$-symbol. Using a standard identity that expresses Turaev–Viro invariants as sums of squared coloured Jones polynomials, they verify the Turaev–Viro volume conjecture for all octahedral fully augmented links without half-twists, and they prove the analogous growth statement for the coloured Jones polynomial. The significance is a diagrammatic, geometry-informed route to a volume result previously obtained by topological and combinatorial surgery arguments.","feed_headline":"Flat octahedral links satisfy the Turaev–Viro volume conjecture","feed_subtitle":"A geometric proof matches their complements to shadow links and fixes the volume at a multiple of the octahedron volume.","key_machinery":"The central object is the nerve of the circle packing associated to a fully augmented link's ideal polyhedral decomposition: a triangulation of $S^2$ whose subdivision pattern detects whether the polyhedra are unions of regular ideal octahedra. The matching quantum machinery is recoupling theory with Jones–Wenzl idempotents, in which triangle pops rewrite the dual graph into a tetrahedral network, each pop contributing one quantum $6j$-symbol, and the final tetrahedron is evaluated through the relation between quantum $6j$-symbols and tetrahedral coefficients. The bridge from link-invariant growth to volume is the identity from the paper's Theorem 4.1, which expresses $TV_r(S^3\\setminus L,q)$ as a sum over colourings of $|J_{L,\\mathbf{i}}(A)|^2$.","core_discovery":"The central claim is Theorem 2.12: for an octahedral fully augmented link $L$ with $c$ crossing circles, the complement $S^3\\setminus L$ is isometric to $\\#_c(S^1\\times S^2)\\setminus \\widetilde{L}$ for some fundamental shadow link $\\widetilde{L}$. The proof starts from the ideal polyhedral decomposition of the fully augmented link complement, reads off the nerve of the associated circle packing, and shows that central subdivisions of the complete graph on four vertices correspond exactly to graph moves on a $D_3$-labelled 4-valent gluing graph; the Borromean family supplies the base case, and each added regular ideal octahedron corresponds to one graph move. On the quantum side, the paper computes the coloured Jones polynomial by recoupling theory and obtains a formula in which the $c-1$ regular ideal octahedra of the decomposition correspond to $c-1$ quantum $6j$-symbols. Combining this with the identity expressing Turaev–Viro invariants as sums of squared coloured Jones polynomials, the paper proves that for odd $r$, $\\lim_{r\\to\\infty} \\frac{2\\pi}{r}\\log |TV_r(S^3\\setminus L, e^{2\\pi i/r})| = 2(c-1)v_8 = \\operatorname{Vol}(S^3\\setminus L)$ whenever $L$ has no half-twists.","pith_inferences":["If the sign uniformity asserted in Lemma 4.15 survives a corrected computation, the only stated obstacle to links with half-twists is cancellation of complex half-twist factors; analytic lower-bound techniques of the type the authors cite may close that gap.","The one-octahedron-per-$6j$-symbol structure suggests that the volume conjecture here is additive: the growth rate decomposes octahedron by octahedron, with each regular ideal octahedron contributing exactly $v_8$ to the logarithmic growth of the invariant.","Because Theorem 2.12 gives isometries rather than homeomorphisms, cusp shapes and Dehn-filling limits of octahedral fully augmented links are readable from the shadow-link side, potentially extending volume-conjecture results to fillings."],"forward_implications":["Every octahedral fully augmented link with $c$ crossing circles has hyperbolic volume exactly $2(c-1)v_8$, realized by the shadow-link model.","The Turaev–Viro volume conjecture holds for every flat octahedral fully augmented link, not only for the previously known examples and families.","The coloured Jones polynomial of such a link is a sum over $c$ summation variables of products of $c$ factors $\\Delta_j\\lambda_{j,a}$, optional half-twist factors, and $c-1$ quantum $6j$-symbols, giving a diagrammatic bookkeeping of the octahedral decomposition.","For flat octahedral fully augmented links, the coloured Jones polynomial evaluated at $t=e^{4\\pi i/(2m+1)}$ grows with rate $2(c-1)v_8$, answering a question raised in the literature for these links."],"supporting_citations":[{"why":"Supplies the growth upper bound for quantum 6j-symbols and the original proof of the Turaev–Viro volume conjecture for fundamental shadow links, the baseline the paper reproves geometrically.","marker":"[2]"},{"why":"Introduces fundamental shadow links and their complements decomposed into regular ideal octahedra, the target model in Theorem 2.12.","marker":"[6]"},{"why":"Gives Theorem 4.1, expressing Turaev–Viro invariants as a sum of squared coloured Jones polynomials, the bridge from skein computations to volume growth.","marker":"[9]"},{"why":"Provides the recoupling-theory formulas (theta nets, tetrahedral networks, triangle pops, and the 6j-symbol formula) used to compute coloured Jones polynomials.","marker":"[13]"},{"why":"Supplies the sign and realness lemma for the relevant quantum 6j-symbols and the asymptotic maximal-6j-symbol result used for the lower bound.","marker":"[16]"},{"why":"Gives the Agol–Thurston polyhedral decomposition of fully augmented link complements that the circle-packing argument starts from.","marker":"[18]"},{"why":"Supplies the lemmas for merging strands and removing crossing circles in the Kauffman bracket, used in the skein-theoretic coloured Jones formula.","marker":"[21]"},{"why":"Gives the circle-packing and nerve characterization of octahedral fully augmented links, including the central-subdivision criterion for when the polyhedra are built of regular octahedra.","marker":"[29]"},{"why":"Provides the change-of-pair and surgery description relating octahedral fully augmented links to fundamental shadow links, which the paper re-derives geometrically.","marker":"[38]"}],"fun_headline_variants":["Geometric proof: octahedral links equal shadow link complements","Octahedral links are shadow links: new volume proof","TV volume conjecture holds for octahedral links via isometry","Flat octahedral links: geometric proof of volume conjecture","Isometry maps octahedral links to fundamental shadow links"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole lower bound rests on Lemma 4.15, which asserts that the sign of the quantum $6j$-symbol $\\begin{Bmatrix} n_r & n_r & j_1\\\\ n_r & n_r & j_2\\end{Bmatrix}$ depends only on $r\\bmod 4$; the proof of that lemma uses equation (4.17), whose denominator appears misprinted, and if the sign uniformity fails then Corollary 4.20 and the lower bound of Lemma 4.25 collapse.","fun_headline_variants_meta":{"raw":{"variants":["Geometric proof: octahedral links equal shadow link complements","Octahedral links are shadow links: new volume proof","TV volume conjecture holds for octahedral links via isometry","Flat octahedral links: geometric proof of volume conjecture","Isometry maps octahedral links to fundamental shadow links"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000721,"raw_usage":{"total_tokens":3286,"prompt_tokens":1048,"completion_tokens":2238,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":664,"completion_tokens_details":{"reasoning_tokens":2155}},"tokens_in":664,"tokens_out":2238,"duration_ms":17837,"temperature":1.0,"reasoning_tokens":2155,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:53:51.063189+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the sign of $S_z^{j_1,j_2}$ in equation (4.17) for a small admissible case such as $r=7$ (so $n_r=2$) with pairs like $(j_1,j_2)=(2,4)$ or $(4,2)$, using the corrected denominator $n_r+(j_1+j_2)/2-z$ in place of the printed $n_r-j_1/2-j_2/2-z$; if the sign depends on $j_1,j_2$ rather than only on $r$, Lemma 4.15 is false. Alternatively, evaluate $|TV_r(S^3\\setminus L,e^{2\\pi i/r})|$ numerically for a small flat octahedral fully augmented link with $c=3$ crossing circles and check whether the limit equals $4v_8$.","supporting_citations":[{"cited_title":"Differential Geom","cited_arxiv_id":null,"evidence_quote":"Supplies the growth upper bound for quantum 6j-symbols and the original proof of the Turaev–Viro volume conjecture for fundamental shadow links, the baseline the paper reproves geometrically."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces fundamental shadow links and their complements decomposed into regular ideal octahedra, the target model in Theorem 2.12."},{"cited_title":"9 (2018), no","cited_arxiv_id":null,"evidence_quote":"Gives Theorem 4.1, expressing Turaev–Viro invariants as a sum of squared coloured Jones polynomials, the bridge from skein computations to volume growth."},{"cited_title":"Kauffman and S´ ostenes L","cited_arxiv_id":null,"evidence_quote":"Provides the recoupling-theory formulas (theta nets, tetrahedral networks, triangle pops, and the 6j-symbol formula) used to compute coloured Jones polynomials."},{"cited_title":"Melby, Asymptotic additivity of the Turaev-Viro invariants for a family of 3-manifolds, J","cited_arxiv_id":null,"evidence_quote":"Supplies the sign and realness lemma for the relevant quantum 6j-symbols and the asymptotic maximal-6j-symbol result used for the lower bound."},{"cited_title":"London Math","cited_arxiv_id":null,"evidence_quote":"Gives the Agol–Thurston polyhedral decomposition of fully augmented link complements that the circle-packing argument starts from."},{"cited_title":"175, Springer-Verlag, New York, 1997","cited_arxiv_id":null,"evidence_quote":"Supplies the lemmas for merging strands and removing crossing circles in the Kauffman bracket, used in the skein-theoretic coloured Jones formula."},{"cited_title":"Math., vol","cited_arxiv_id":null,"evidence_quote":"Gives the circle-packing and nerve characterization of octahedral fully augmented links, including the central-subdivision criterion for when the polyhedra are built of regular octahedra."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the change-of-pair and surgery description relating octahedral fully augmented links to fundamental shadow links, which the paper re-derives geometrically."}],"review_version":1}