{"id":"850cfee3-1568-4bea-80a7-6c55922b62f0","arxiv_id":"1908.05382","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors construct infinite families of knots and virtual knots where any two members are one region crossing change or arc shift move apart, proving arbitrarily high-dimensional simplexes exist in both Gordian complexes.","lead":"This paper shows that the Gordian complex of knots under region crossing changes and the Gordian complex of virtual knots under arc shift moves both contain simplexes of every dimension. The authors build infinite families in which any two knots differ by exactly one such move, and they compute polynomial invariants that the virtual family shares.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 3.4 relies on virtual connected sum L♯VK_i without addressing non-well-definedness of the operation; the paper neither fixes representatives nor proves the resulting vertices are distinct.","rationale":"The reader's weakest_assumption identifies exactly the gap I consider most load-bearing: the transition from the explicit infinite family VK_n to an arbitrary virtual knot L via connected sum in Corollary 3.4. The vertex set of G_A is the set of virtual knot equivalence classes, and virtual connected sum is known not to be a well-defined operation on those classes. The paper neither cites this subtlety nor supplies the needed well-definedness or a diagram-level repair. The same objection does not apply to the classical Theorem 2.3, where connected sum is well-defined, and the Kawauchi-polynomial distinctness proof is much more complete. I do not see a more serious flaw in the main constructions: the tangle-matrix computation of f-polynomial degrees in Section 4 is a concrete algebraic recurrence that appears internally consistent, and Proposition 5.1 is a useful non-distinguishing check. The gap is specific and likely repairable by replacing Corollary 3.4 with a statement about long or based virtual knots, or by fixing representatives and proving the f-polynomial degree is additive for these connected sums. Since this is exactly the reader's conditional concern, no change to the reader's verdict is needed.","tokens_in":15641,"tokens_out":17077,"duration_ms":178785,"concrete_test":"Fix a nontrivial virtual knot L, for example the virtual trefoil 2.1, and choose two diagrams of L that are related by generalized Reidemeister moves but allow connected sums at different base arcs. Form D_1 = L#VK_1 and D_2 = L'#VK_1, where the connected sum is taken at different arcs. Compute a strong invariant of both resulting virtual knots, such as the two-variable F^2 polynomial used in the paper's Table 1 or the writhe-normalized Kauffman f-polynomial. If the invariants differ, the vertex 'L♯VK_1' is not well-defined and Corollary 3.4 as stated does not follow; if the invariants agree for a systematic family of choices, that would suggest the proof can be repaired by fixing based representatives.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's advertised virtual result, Corollary 3.4, is proved in one sentence: take the family {L♯VK_0, …, L♯VK_n}. This step is load-bearing because G_A is defined on virtual knot isotopy classes, while connected sum of virtual knots is not generally well-defined on equivalence classes: the result can depend on the chosen diagrams and on the arcs where the summands are joined. The proof gives no reference, no use of long or based virtual knots, and no argument that the specific sums used are independent of these choices. Even reading the construction as fixing a diagram of L and a basepoint, the proof never shows that the new vertices L♯VK_i are pairwise distinct; the f-polynomial degree computation in Section 4 applies only to the unsunned VK_n and is not automatically transferred to connected sums with an arbitrary L. Since the abstract explicitly promises an arbitrarily high-dimensional simplex containing any given virtual knot diagram, this gap affects the central claim. The analogous classical connected-sum step in Theorem 2.3 is harmless because classical connected sum is well-defined on isotopy classes, but the virtual case is not the same.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies two Gordian complexes: the complex of classical knots under region crossing change (r.c.c.) and the complex of virtual knots under arc shift moves. It constructs, for each n, an n-simplex in the r.c.c. Gordian complex containing a prescribed classical knot (Theorem 2.3), and an infinite family of virtual knots that are pairwise at arc-shift distance one (Theorem 3.3). It then claims, in Corollary 3.4, that any virtual knot L is a vertex of an arbitrarily high-dimensional simplex by taking connected sums with the constructed family. Distinctness of the classical family is argued via Kawauchi's coefficient polynomials, and distinctness of the virtual family via maximal degrees of the Kauffman f-polynomial; the paper also computes the affine index polynomial of the virtual knots in Proposition 5.1.","tokens_in":15782,"tokens_out":10227,"duration_ms":100954,"significance":"The constructive approach is concrete and potentially useful, giving explicit infinite cliques in two Gordian complexes. If the virtual-knot extension were proved, the result would nicely parallel known results for the v-move and forbidden-move Gordian complexes. The invariant computations are a genuine strength: Kawauchi's c0 polynomial and the f-polynomial degree method are applied to explicit families rather than asserted abstractly. However, the advertised generalization to an arbitrary virtual knot is not established, because it depends on connected sums of virtual knots without addressing well-definedness or distinctness. The arithmetic error in the degree computation and the unproved cancellation assumption in the max-degree induction are further load-bearing gaps. The unknot-based results are plausible and likely repairable; the arbitrary-virtual-knot claim requires a substantially different argument.","major_comments":[{"comment":"The proof of Corollary 3.4 is a single sentence asserting that the family {L♯VK_0, ..., L♯VK_n} is the desired simplex. This is not justified. Connected sum of virtual knots is not a well-defined operation on virtual knot isotopy classes: the result generally depends on the chosen diagrams and on the arcs at which the summands are joined. The paper neither fixes representatives nor cites a reference making the operation well-defined in a restricted setting (e.g., long or based virtual knots). Moreover, the proof never shows that the vertices L♯VK_i are pairwise distinct; the f-polynomial degree computation in the proof of Theorem 3.3 applies only to the unsunned VK_n and does not automatically transfer to connected sums with an arbitrary L. Since the abstract explicitly promises an arbitrarily high-dimensional simplex containing any given virtual knot, this gap directly affects the paper's central virtual claim.","section":"Section 4, proof of Corollary 3.4"},{"comment":"The computation of the maximum degree of the bracket polynomial of VK_n contains a concrete arithmetic error. Equation (10) states that the max-degree row vector (4,−2,0) times (10n+2, 6n, 10n) gives max degree 10n+6, but the left-hand side evaluates to 4(10n+2) − 2(6n) = 28n+8. This invalidates the stated conclusion that the highest power of the f-polynomial is 4n. Additionally, the 'max' notation used in Proposition 4.2 treats the degree of a sum of products as the maximum of the degrees of the summands; this is an upper bound and can fail by cancellation of leading terms. The proof supplies no check that the relevant leading coefficients do not cancel. The distinctness of the VK_n may be salvageable by a corrected degree computation, but as written the proof is not sound.","section":"Section 4, Eq. (10) and Proposition 4.2"},{"comment":"The reduction from an arbitrary knot K0 to the unknot is stated in one sentence: 'The general statement will follow by taking connected sums.' For classical knots this can be justified, but the manuscript should spell out the two needed facts: (i) if K_i and K_j differ by one r.c.c. in a region, then K0♯K_i and K0♯K_j differ by one r.c.c. in the corresponding region away from the connected-sum tube, and (ii) the knots K0♯K_i are distinct, which follows from the uniqueness of prime decomposition for classical knots. Without these details, the proof of the advertised 'any knot K0' statement is incomplete, even though the gaps are standard.","section":"Section 2, proof of Theorem 2.3"}],"minor_comments":[{"comment":"The manuscript contains many typos and grammatical slips, including 'the exists' in Theorem 2.3, 'bases' in the introduction, 'skien triple' in Section 2, 'Kauffman' for 'Kauffman' in several places, and 'chamge' near the end of Section 2. A careful proofreading pass is needed.","section":"Throughout"},{"comment":"The F-polynomial computations are presented only for n = 1,...,10, and the statement that the polynomials are 'the same as for the case n = 3' for n = 4,...,10 is empirical, not a proof for all n. If this discussion is retained, it should be labeled as a finite computation or supplemented with a general argument.","section":"Section 5, Table 1"},{"comment":"The claim that applying one arc shift move in block c_j of VK_n yields the diagram VK_{j-1} is asserted by inspection of Figures 20 and 21. A more explicit description of the simplification sequence would help the reader verify this key geometric step.","section":"Section 4, proof of Theorem 3.3"},{"comment":"The definition of the r.c.c. distance is worded confusingly: 'Minimum no. of r.c.c. required to convert all such diagrams D into D'' suggests a minimum over a collection of diagrams without a fully formal quantifier over diagrams of K and K'. A precise definition would improve readability.","section":"Section 2, definition of d_R"}],"recommendation":"major_revision","confidential_remarks":"I recommend major revision rather than rejection. The explicit constructions and invariant computations are valuable, and the flaws appear local in spirit: a clear arithmetic slip in Eq. (10), an unproved cancellation assumption in the max-degree induction, and a missing treatment of virtual connected sum in Corollary 3.4. However, the abstract and Corollary 3.4 currently overclaim; the authors should either restrict the virtual result to the unknot or to a setting where connected sum is well-defined (e.g., long or based virtual knots), or provide a genuine proof for arbitrary L. The arithmetic inconsistency in Eq. (10) must be corrected before the distinctness argument can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it so you know the construction, but don't trust the corollary.\n\nThe paper does something concrete: for the region crossing change on classical knots, it builds explicit knots K_m that are pairwise one r.c.c. apart, and proves they are distinct via Kawauchi's c_0 polynomial. That part is solid. The same template is adapted to virtual knots with the arc shift move, giving VK_n pairwise one arc shift apart, distinguished by the maximum degree of Kauffman's f-polynomial via a tangle-matrix recurrence. That is also plausible; the recurrence is explicit, though there are typos in the printed B matrix and a garbled sign sentence in the affine index section.\n\nThe soft spot is Corollary 3.4. The proof is one sentence: connect-sum L with each VK_i. But connected sum of virtual knots is not well-defined on equivalence classes, and the paper neither fixes long or based representatives nor shows the resulting L#VK_i are pairwise distinct. The f-polynomial degree calculation does not transfer, because the f-polynomial of a virtual connected sum is not a simple product when the summand is not classical. Since the abstract advertises arbitrarily high-dimensional simplexes containing any given virtual knot, this gap has to be fixed before publication. It is fixable, but it is not a presentation issue.\n\nMinor quibbles: the max-degree induction for the tangle matrix assumes the degree of a sum of rational functions is the max of the degrees, which can fail by cancellation; the authors should check that the explicit entries do not cancel. The classical connected-sum step in Theorem 2.3 is harmless because classical connected sum is well-defined.\n\nBottom line: worth a serious referee, but the verdict should be 'revise.' If the authors replace Corollary 3.4 with a statement about long virtual knots, or construct a family directly around L, the paper would be a useful addition to the compendium of Gordian complexes.","headline":"Useful explicit constructions for two Gordian complexes, but the virtual generalization to arbitrary vertices rests on an unjustified connected sum.","tokens_in":16377,"tokens_out":3783,"would_cite":false,"duration_ms":38093,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57M25","57M27"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper builds infinite families of knots, and of virtual knots, whose members are pairwise one local move apart, producing n-simplices for every n in both Gordian complexes.","keywords":["Gordian complex","region crossing change","arc shift move","virtual knots","simplicial complex","unknotting operation","f-polynomial","affine index polynomial"],"falsifier":"Take a nontrivial virtual knot $L$ and form $L\\sharp VK_1$ using two different choices of connected-sum diagrams or summation arcs; if the two results are inequivalent virtual knots, then the asserted general-vertex statement does not follow from the given construction.","tokens_in":15364,"feed_emoji":"🪢","tokens_out":11696,"duration_ms":102423,"temperature":0.7,"pith_summary":"The paper builds, for each base knot, an infinite family of knots in which every two members are changed into one another by a single region crossing change; and for each base virtual knot, an infinite family of virtual knots in which every two members are changed into one another by a single arc shift. Because every pair is distance one, each family spans an $n$-simplex in the corresponding Gordian complex for arbitrarily large $n$. The paper proves this directly for every classical knot, and asserts the same for every virtual knot via connected sums with the constructed family. Distinctness is certified by polynomial invariants: the $c_0$ coefficient polynomial for the classical knots and the maximum degree of the $f$-polynomial for the virtual knots. All constructed virtual knots share the same affine index polynomial, so that invariant cannot separate them.","feed_headline":"Every knot and virtual knot anchors simplices of every dimension","feed_subtitle":"Region crossing changes and arc shifts both produce infinite families of knots that are pairwise one move apart.","key_machinery":"The load-bearing object is the block diagram family together with the invariants that certify distinctness. In the classical case $K_m$ has $m$ copies of one block; region crossing change at the $i$-th region deletes blocks, giving $K_{i-1}$. In the virtual case $VK_n$ has blocks $c_0,\\dots,c_n$ made of alternating classical and virtual crossings on a two-strand braid; an arc shift in block $c_j$ frees that braid and cancels all crossings to the right, giving $VK_{j-1}$. Distinctness of $K_m$ is shown through the $c_0$ coefficient polynomial of the skein polynomial, whose degree increases with $m$. Distinctness of $VK_n$ is shown through the bracket polynomial and the normalized $f$-polynomial: a $3\\times 3$ transfer matrix tracks the $D$-, $N$-, and $X$-closures of the repeated block, and induction gives maximum $f$-degree $4n$. The paper also labels arcs in $VK_n$ to compute the affine index polynomial, obtaining the common value $2 - t^2 - t^{-2}$ for every $n$.","core_discovery":"The central discovery is a 'one block per move' diagram family. The knot $K_m$ is built from $m$ identical blocks, and applying a region crossing change at the $i$-th region of $K_m$ returns $K_{i-1}$; hence $K_i$ and $K_j$ are at region-crossing-change distance one for every pair. The virtual knot $VK_n$ is built from $n+1$ blocks, and one arc shift applied to a two-strand braid in block $c_j$ collapses all crossings to its right under the standard diagram moves, returning $VK_{j-1}$; hence all $VK_n$ sit at arc-shift distance one from each other. To know these really are distinct vertices, the paper computes the $c_0$ polynomial for $K_m$, whose maximum degree grows with $m$, and the $f$-polynomial for $VK_n$, whose maximum degree is $4n$. The argument then extends from the unknot to arbitrary knots and virtual knots by connected sum, so every $0$-simplex is contained in an $n$-simplex for every positive $n$.","pith_inferences":["If the virtual connected-sum step can be made well-defined, for instance by using long virtual knots, the arc-shift result would hold for every virtual knot; without such a repair the general-vertex statement is not established by the paper's proof.","The same block-collapse recipe may transfer to other unknotting operations that act locally and kill an entire block, yielding arbitrary-dimensional simplices in their Gordian complexes.","Because all $VK_n$ share one affine index polynomial, it would be informative to test stronger invariants, such as the odd writhe or two-variable $F$-polynomials beyond the range $n \\le 10$ checked here.","The paper's boundedness observation for region crossing change suggests that, unlike crossing-change Gordian complexes, high-dimensional simplices here do not imply long geodesic paths; the geometry is concentrated in a two-tier structure."],"forward_implications":["For every classical knot $K_0$, the region-crossing-change Gordian complex contains an $n$-simplex with vertex $K_0$ for every positive integer $n$.","For every virtual knot $L$, the arc-shift Gordian complex is asserted to contain an $n$-simplex with vertex $L$ for every positive integer $n$.","The constructed families form infinite cliques in the distance-one graph of each move: infinitely many pairwise distinct knots that are all one move apart.","Because the region-crossing-change distance between any two classical knots is at most two, the arbitrarily high-dimensional simplices live inside a Gordian complex of diameter at most two.","The common affine index polynomial $2 - t^2 - t^{-2}$ means the arc-shift construction is invisible to that invariant, so it cannot be used to separate members of the family."],"supporting_citations":[{"why":"gives the original Gordian complex definition and the block-family method this paper extends.","marker":"[6]"},{"why":"proves region crossing change is an unknotting operation, making the r.c.c. distance finite.","marker":"[14]"},{"why":"supplies the $c_0$ coefficient polynomial used to prove the constructed classical knots are distinct.","marker":"[7]"},{"why":"supplies the tangle-state transfer-matrix method and the virtual family modified for arc shifts.","marker":"[3]"},{"why":"establishes arc shift as an unknotting operation for virtual knots.","marker":"[2]"},{"why":"introduces virtual knots and the bracket polynomial behind the $f$-polynomial computation.","marker":"[9]"},{"why":"defines the affine index polynomial computed for the constructed virtual knots.","marker":"[10]"}],"fun_headline_variants":["Infinite knot families: one move apart, every dimension","Knot simplices of any dimension from single-move families","Region flips and arc shifts build high-dimensional knot complexes","Every knot lies in infinite family of one-move neighbors","One move apart: knots form simplices of every dimension"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument that the arbitrary-simplex statement passes from the unknot to any virtual knot $L$ assumes that the connected sum $L\\sharp VK_i$ is a well-defined virtual knot, independent of chosen diagrams and summation arcs; virtual connected sum is not generally well-defined, and the paper supplies no justification for this step.","fun_headline_variants_meta":{"raw":{"variants":["Infinite knot families: one move apart, every dimension","Knot simplices of any dimension from single-move families","Region flips and arc shifts build high-dimensional knot complexes","Every knot lies in infinite family of one-move neighbors","One move apart: knots form simplices of every dimension"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000711,"raw_usage":{"total_tokens":3208,"prompt_tokens":963,"completion_tokens":2245,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":579,"completion_tokens_details":{"reasoning_tokens":2163}},"tokens_in":579,"tokens_out":2245,"duration_ms":13467,"temperature":1.0,"reasoning_tokens":2163,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:16:11.534411+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a nontrivial virtual knot $L$ and form $L\\sharp VK_1$ using two different choices of connected-sum diagrams or summation arcs; if the two results are inequivalent virtual knots, then the asserted general-vertex statement does not follow from the given construction.","supporting_citations":[{"cited_title":"Hirasawa, Y","cited_arxiv_id":null,"evidence_quote":"gives the original Gordian complex definition and the block-family method this paper extends."},{"cited_title":"Shimizu, Region crossing change is an unknotting operation,J","cited_arxiv_id":null,"evidence_quote":"proves region crossing change is an unknotting operation, making the r.c.c. distance finite."},{"cited_title":"Kawauchi, On coefficient polynomials of the skein polynomial of an oriented link, Kobe J","cited_arxiv_id":null,"evidence_quote":"supplies the $c_0$ coefficient polynomial used to prove the constructed classical knots are distinct."},{"cited_title":"Horiuchi, K","cited_arxiv_id":null,"evidence_quote":"supplies the tangle-state transfer-matrix method and the virtual family modified for arc shifts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes arc shift as an unknotting operation for virtual knots."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces virtual knots and the bracket polynomial behind the $f$-polynomial computation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines the affine index polynomial computed for the constructed virtual knots."}],"review_version":1}