Weinstein (arXiv:2511.18016, 2025) proved that if a diagram is obtained from a composite knot diagram K0#K1K_0 \# K_1 by a restricted set of Reidemeister moves - classical moves performed away from a distinguished critical point of the diagram, plus prescribed sliding moves of the two partial arcs across that point - then its crossing count is at least c(K0)+c(K1)c(K_0) + c(K_1), where cc is the minimal crossing number. The proof splits crossings into non-hybrid ones, bounded via the minimal ray intersection number MINray, and hybrid ones, bounded by a separate combinatorial argument; both bounds are tied to the restricted move sets and fail for ordinary, unrestricted Reidemeister equivalence. The paper poses two open questions, recorded here as one linked problem. (1) Find invariants T±(K)\mathcal{T}^{\pm}(K) of knot diagrams that make both bounds provable under full Reidemeister equivalence - which, substituted into the paper's proof template, would establish the classical additivity conjecture c(K1#K2)=c(K1)+c(K2)c(K_1 \# K_2) = c(K_1) + c(K_2) for all knots. (2) Find a less ad hoc, composition-stable topological interpretation of the restricted-move additivity phenomenon. No literature answers either question. A very recent unrefereed preprint (Kawauchi, arXiv:2608.13789, 13 Aug 2026) claims a full proof of additivity by unrelated 4-dimensional methods, but it is unverified and does not address these questions.

Background

A knot diagram is a drawing of a knot (an embedded closed curve in 3-dimensional space) as a generic immersed closed curve in the plane, with over/under information recorded at each double point; two diagrams present the same knot exactly when they are connected by the three local Reidemeister moves R1 (twist or untwist a kink), R2 (slide one strand past another, creating or deleting a pair of crossings), and R3 (slide a strand across a crossing). The crossing number c(K)c(K) of a knot KK is the minimum number of crossings over all diagrams of KK; it is the knot's fundamental complexity measure. The connected sum K0#K1K_0 \# K_1 of two knots is formed by cutting each knot open and gluing the ends together in pairs; a knot is composite if it arises this way from two nontrivial knots. Stacking minimal diagrams gives c(K0#K1)≤c(K0)+c(K1)c(K_0 \# K_1) \le c(K_0) + c(K_1), and the additivity conjecture asserts that equality always holds. The conjecture is decades old and open in general. Equality is known when both factors are alternating knots (knots admitting a diagram in which over- and under-crossings alternate along the strand), by Kauffman's state-model analysis of the Jones polynomial [Kauffman 1987], and when both factors are torus knots (knots lying on the surface of an unknotted torus in 3-space) [Diao 2004]; the best general result is Lackenby's linear inequality c(K1)+⋯+c(Kn)≤152 c(K1#⋯#Kn)c(K_1) + \cdots + c(K_n) \le 152\, c(K_1 \# \cdots \# K_n) [Lackenby 2009].

Weinstein [2025] approaches the conjecture through restricted equivalence relations on diagrams. A centered knot diagram is a diagram drawn as a generic immersed arc in the unit square □\square, running from the bottom midpoint to the top midpoint and passing through the center o=(1/2,1/2)o = (1/2, 1/2), called the critical point; its restrictions K+K^{+} and K−K^{-} to the upper and lower halves of the square are the upper and lower partial diagrams. Besides the usual Reidemeister moves performed in disks that avoid oo (together with ambient deformations of the square fixing its boundary and the point oo), one may allow sliding moves, in which a subarc of K+K^{+} or K−K^{-} slides over or under the critical point. Fixing which slides are allowed yields five move sets; for example Ropp1∗R^{*}_{opp1} consists of the non-sliding moves together with the two slides in which the upper arc passes over oo and the lower arc passes under oo. The induced equivalence relations on diagrams are denoted ∼RX\sim^{X}_{R} for X∈{∗,o1,o2,s1,s2}X \in \{*, o1, o2, s1, s2\}, with X=∗X = * the most restrictive (no slides). Theorem 1 of the paper states that if K^∼RXK\hat{K} \sim^{X}_{R} K where KK is a composite centered diagram K0#K1K_0 \# K_1, then the crossing count χ(K^)\chi(\hat{K}) satisfies χ(K^)≥c(K0)+c(K1)\chi(\hat{K}) \ge c(K_0) + c(K_1): additivity holds within each restricted equivalence class.

The proof decomposes the crossings of K^\hat{K} into non-hybrid crossings (both strands within K^+\hat{K}^{+}, or both within K^−\hat{K}^{-}) and hybrid crossings (one strand from each partial diagram). The non-hybrid crossings are bounded by Corollary 5: χ(K^+)+MINray(K^+)≥c(K1)\chi(\hat{K}^{+}) + \mathrm{MIN}_{\mathrm{ray}}(\hat{K}^{+}) \ge c(K_1) and χ(K^−)+MINray(K^−)≥c(K0)\chi(\hat{K}^{-}) + \mathrm{MIN}_{\mathrm{ray}}(\hat{K}^{-}) \ge c(K_0), where MINray(T)\mathrm{MIN}_{\mathrm{ray}}(T) is the minimal ray intersection number, i.e. the minimum, over all rays JJ (simple curves in □\square running from the critical point oo to the boundary), of the number of transverse intersection points between TT and JJ. The hybrid crossings are bounded by Corollary 7 and Proposition 7.1: i(K^−,K^+)≥MINray(K^−)+MINray(K^+)i(\hat{K}^{-}, \hat{K}^{+}) \ge \mathrm{MIN}_{\mathrm{ray}}(\hat{K}^{-}) + \mathrm{MIN}_{\mathrm{ray}}(\hat{K}^{+}), where i(A,B)i(A, B) denotes the number of transverse intersection points between AA and BB; this is proved via an elaborate fan deformation of the diagram and a graph-theoretic counting lemma. The obstruction to running the same argument under ordinary, unrestricted Reidemeister equivalence ∼R\sim_{R} concerns the moves through the critical point: MINray\mathrm{MIN}_{\mathrm{ray}} is proved invariant only under the ambient move R0 (Lemma 5.2), and the proof of Corollary 5 (Proposition 5.1) closes the partial diagrams into full knot diagrams along a ray run through a tube around oo that is fixed by the ambient homeomorphism, which works only because the restricted move sets control which side of oo the arcs slide on (for X=o1X = o1, the subarcs of K^+\hat{K}^{+} are never allowed to slide under oo); unrestricted Reidemeister equivalence permits arbitrary passages across oo in both directions and breaks this construction, so both bounds are tied to the restricted relations. The paper also gives a topological reading of the X=o1X = o1 case (Corollary 2): the crossing number cannot drop below c(K0)+c(K1)c(K_0) + c(K_1) under any self-homeomorphism of the cube □×I\square \times I satisfying a certain condition relative to a diagonal cross-section of the cube; the author concedes (Remark 3.1) that this characterization is ad hoc and that the class of homeomorphisms satisfying it is not closed under composition, so no natural mapping-class picture (a description through equivalence classes of homeomorphisms forming a group under composition) of the phenomenon is known. Section 10 of the paper poses the two questions recorded here.

Problem Statement

Let K0K_0, K1K_1 be knots, let K=K0#K1K = K_0 \# K_1 be a composite centered knot diagram, and let ∼R\sim_{R} denote equivalence of diagrams under unrestricted Reidemeister moves. The problem is the pair of questions posed in Section 10 of Weinstein (arXiv:2511.18016), whose common objective is to convert the restricted-move proof of crossing-number additivity into an unrestricted one and to understand, geometrically, what the restriction means.

(1) The invariants question (Question 2 of the paper). Construct invariants T−(K)\mathcal{T}^{-}(K) and T+(K)\mathcal{T}^{+}(K) of knot diagrams such that:

(i) the non-hybrid bound of Corollary 5 holds with T±\mathcal{T}^{\pm} in place of MINray(K±)\mathrm{MIN}_{\mathrm{ray}}(K^{\pm}) and with ∼R\sim_{R} in place of the restricted ∼RX\sim^{X}_{R} - that is, for every composite centered diagram K=K0#K1K = K_0 \# K_1 and every K^∼RK\hat{K} \sim_{R} K, evaluating the invariants on the diagram under consideration, χ(K^+)+T+(K^)≥c(K1)andχ(K^−)+T−(K^)≥c(K0);\chi(\hat{K}^{+}) + \mathcal{T}^{+}(\hat{K}) \ge c(K_1) \quad \text{and} \quad \chi(\hat{K}^{-}) + \mathcal{T}^{-}(\hat{K}) \ge c(K_0);

(ii) the hybrid-crossing bound of Proposition 7.1 still holds with the same substitution - for every centered subdivided tangle diagram TT (a tangle diagram generalizes knot and link diagrams to a generic immersed one-manifold in the square with over/under data at the double points, possibly with several strands whose endpoints lie on the boundary of the square and with closed loop components; a subdivided t-diagram is one with a distinguished strand passing through the critical point oo, its upper and lower partial diagrams T+T^{+} and T−T^{-} being defined as for knot diagrams), i(T−,T+)≥T−(T)+T+(T),i(T^{-}, T^{+}) \ge \mathcal{T}^{-}(T) + \mathcal{T}^{+}(T), where i(A,B)i(A, B) denotes the number of transverse intersection points between AA and BB.

As the author observes, invariants with these two properties, substituted into the proof template of Theorem 1, would establish the additivity conjecture c(K1#K2)=c(K1)+c(K2)c(K_1 \# K_2) = c(K_1) + c(K_2) for all knots under ordinary Reidemeister equivalence.

(2) The interpretation question (Question 1 of the paper). Give a topological interpretation, less ad hoc than the diagonal cross-section condition of Corollary 2 and stable under composition, of the fact (the X=o1X = o1 case of Theorem 1) that the moves in Ropp1∗R^{*}_{opp1} cannot reduce the number of crossings of a composite knot below c(K0)+c(K1)c(K_0) + c(K_1).

The two parts are linked: part (1) is the load-bearing completion of the paper's proof strategy, and part (2) asks for the geometric reason the strategy currently needs the restriction.

References

  1. Vadim Weinstein, Additivity of Crossing Number via Restricted Reidemeister Moves, arXiv:2511.18016 (2025), https://arxiv.org/abs/2511.18016
  2. Marc Lackenby, The crossing number of composite knots, Journal of Topology 2(4): 747-768 (2009), DOI 10.1112/jtopol/jtp028, https://doi.org/10.1112/jtopol/jtp028 (preprint arXiv:0805.4706, https://arxiv.org/abs/0805.4706)
  3. Louis H. Kauffman, State models and the Jones polynomial, Topology 26: 395-407 (1987), DOI 10.1016/0040-9383(87)90009-7, https://doi.org/10.1016/0040-9383(87)90009-7
  4. Yuanan Diao, The additivity of crossing numbers, Journal of Knot Theory and Its Ramifications 13: 857-866 (2004), DOI 10.1142/S0218216504003524, https://doi.org/10.1142/S0218216504003524
  5. Akio Kawauchi, The crossing numbers of knots and links via 4D topology, arXiv:2608.13789 (2026), https://arxiv.org/abs/2608.13789
  6. Akio Kawauchi, Fusion of boundary surface-link and ribbonness, arXiv:2507.18154 (2025, last revised 2026), https://arxiv.org/abs/2507.18154