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arxiv: 2601.11481 · v2 · submitted 2026-01-16 · 🧮 math.GT · cond-mat.soft· cond-mat.stat-mech

Entanglement complexity of spanning pairs of lattice polygons

Ryan Blair , Puttipong Pongtanapaisan , Christine E. Soteros This is my paper

Pith reviewed 2026-05-16 13:44 UTC · model grok-4.3

classification 🧮 math.GT cond-mat.softcond-mat.stat-mech
keywords spanning linkslattice polygonsentanglement complexitytangle productsgood measuresbridge numbersplitting numberlinking probability
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The pith

All but exponentially few large spanning pairs of lattice polygons have entanglement complexity that grows at least linearly with size.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the notion of a good measure of knot complexity to a measure F for spanning links formed by two self-avoiding polygons that together span a lattice tube. It uses tangle products to show that F increases when any component knot becomes more complex in its prime decomposition. The central result establishes that for size-m 2SAPs, all but an exponentially small fraction have F at least linear in m. This builds on earlier proofs that such pairs are almost surely linked and supplies a quantitative lower bound on their entanglement.

Core claim

Good measures of knot complexity extend via tangle products to good measures F of spanning-link complexity for k-component links. For any such F, all but exponentially few size-m 2SAPs satisfy that F grows at least linearly in m as m tends to infinity. Classical link invariants including bridge number and splitting number yield good measures under the given definition.

What carries the argument

The good measure F of spanning-link complexity, constructed from tangle products that combine knot components while preserving monotonicity under increasing knot complexity.

If this is right

  • More complex prime knot factors on either component strictly raise the F value of the 2SAP.
  • Any good measure of knot complexity automatically supplies a good measure of spanning-link complexity.
  • Invariants such as bridge number and splitting number become good measures for 2SAPs.
  • Certain two-component links appear as 2SAPs only when at least one component is forced into a non-minimal bridge-number embedding.
  • Tube cross-section dimensions determine both the embeddable link types and the geometric constraints on their possible F values.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The linear lower bound on F supplies a concrete rate at which entanglement accumulates in confined two-ring polymer models beyond the mere fact of linking.
  • Numerical sampling of 2SAPs for moderate m could test whether the exponential decay of low-F examples already appears at accessible sizes.
  • The same tangle-product technique may apply directly to spanning links with three or more components once suitable lattice models are defined.

Load-bearing premise

The tangle-product construction that turns good knot measures into good spanning-link measures preserves the required increasing properties with respect to component knot complexity.

What would settle it

An explicit enumeration or Monte-Carlo sampling for successively larger m that finds a positive-density subset of 2SAPs whose F remains bounded by a fixed constant.

Figures

Figures reproduced from arXiv: 2601.11481 by Christine E. Soteros, Puttipong Pongtanapaisan, Ryan Blair.

Figure 1
Figure 1. Figure 1: (a) An embedding of a 2-component unlink in a (2 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Two 2-component links (top) are concatenated together by component-wise connected sum operations [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) The link L(K) is the split union of k knots (shown in black for k = 2) where one is knot-type K (localized in the dashed blue circle) and the others are the unknot (01). L(K) is also used to represent the k-tangle triple (L(K), G1, G2) where G1 and G2 are the k-star graphs shown in green and red respectively for the case k = 2 (see Definition 5 in Section 2.3) (b) A pattern that increases complexities … view at source ↗
Figure 4
Figure 4. Figure 4: (a) Two 3-tangle-triples (L(K1 ), G1 1, G1 2) and (L(K2 ), G2 1, G2 2) where Ki for i = 1, 2 is any knot. (b) A result of performing a 3-strand tangle product to get (L(K1#K2 ), G1 1, G2 2). 1. if U is an unlink, F(U) = 0, 2. if a link L = L ′ ∪ U is a split link with splitting sphere separating L ′ from U, then F(L) = F(L ′ ), 3. the restriction F : L 1 → [0,∞) is a good measure of knot complexity, and 4.… view at source ↗
Figure 5
Figure 5. Figure 5: (a) The number of prime knot factors may decrease under concatenation. The resulting link after concate [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A family of 2-component links that shows that the number of [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The link L6a1 is tricolorable This coloring idea can be generalized from using integers modulo p to using elements of any group. More precisely, one can show that a link is tricolorable if there exists a surjective homomorphism from the fundamental group of the link exterior πL to the 6 element dihedral group. Similarly, we can obtain Col3(L) by counting the number of homomorphisms from πL to the 6 element… view at source ↗
Figure 8
Figure 8. Figure 8: A tangle obtained from a family of 2-tangle-triples that can be used to show that the number of incom [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A 2SAP with an (M + 1)(N + 1) section. Ha Hb [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: After reductions, a nontrivial 2SAP does not contain an ( [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Cutting the special arc colored in blue at its ends gives a local pattern. The rectangles represent braid [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: A case where trying to create a local pattern may need a larger tube size. The rectangles represent braid [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: A satellite 2-component link is formed by taking two solid tori each containing a knot, and then knotting [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: A nested saddle. 3.2.3 Proof of the main theorem In this section we let L = K1 ∪ K2 be a 2-component satellite link with patterns (Vc1, Kc1),(Vc2, Kc2) and companions L 0 1 , L0 2 . Recall from Section 3.2.1 that Vi is the image of Vc1 in S 3 . We additionally assume that K1 and L 0 1 are unknots. In particular, the dual index ω1 of L 0 1 exists. Given a space A, we will let |A| denote the number of conne… view at source ↗
Figure 15
Figure 15. Figure 15: (Left) A Type I saddle. (Right) A Type II saddle. [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: If c1 and c3 each bounds a disk in ∂V1, then c2 does also. subject to the constraint that the K1 component of ℓ ′ has j maxima. Let σ be the highest essential saddle. Then, σ is nested with respect to V1. Proof. Let S be the height of σ = s1 ∨ s2. Let Di (i = 1, 2) be the level disks in h −1 (S) bounded by si such that int(D1) ∩ s2 = ∅ and int(D2) ∩ s1 = ∅. Suppose that σ is not nested with respect to V1.… view at source ↗
Figure 17
Figure 17. Figure 17: A knee component. Lemma 7. Fix j ≥ β(K1), where β(K1) is the bridge number of K1 in S 3 . Suppose that the index σ1 of (Vc1, Kc1) is strictly greater than j. Additionally, suppose that ℓ ′ is an EH-trunk minimizing embedding of L subject to the constraint that the K1 component of ℓ ′ has j maxima. Let the highest essential saddle of FV1 lie in the level sphere h −1 (r). Then, |L ∩ h −1 (r)| ≥ 2ω1σ2 + 2. P… view at source ↗
Figure 18
Figure 18. Figure 18: Two morse embeddings of the same link type. By the proof of Theorem 7, the top embedding can be [PITH_FULL_IMAGE:figures/full_fig_p023_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Two morse embeddings of the same link type. By the proof of Theorem 7, the top embedding can be [PITH_FULL_IMAGE:figures/full_fig_p023_19.png] view at source ↗
read the original abstract

We study the entanglement complexity of a system consisting of two simple-closed curves (self-avoiding polygons) that span a lattice tube, referred to as a 2SAP. 2SAPs are of interest as the first known model of confined ring polymers where the linking probability goes to 1 exponentially with the size of the system. Atapour et al proved this in 2010 by showing that all but exponentially few sufficiently large 2SAPs contain a pattern that guarantees the 2SAP is non-split, provided that the requisite pattern fits in the tube. This result was recently extended to all tubes sizes that admit non-trivial links. Here we develop and apply knot theory results to answer more general questions about the entanglement complexity of 2SAPs. We first extend the 1992 concept of a good measure of knot complexity to a good measure, $F$, of spanning-link complexity for $k$-component links. Using tangle products, we show, for example, that the more complex the prime knot decomposition of any component of a given link type, the greater its $F$-measure. We then prove that all but exponentially few size $m$ 2SAPs have $F$ complexity that grows at least linearly in $m$ as $m\to \infty$. We establish that good measures of knot complexity yield good measures of spanning-link complexity. We also establish conditions whereby more general link invariants can yield good measures. In particular, we establish that measures based on several classical invariants are good measures by our definition, eg bridge number or the splitting number. Finally, we consider how the tube dimensions affect which links are embeddable as 2SAPs as well as geometric restrictions on the entanglement complexity of the embeddings. For example, we establish that there are two-component links that occur as 2SAPs in a given tube size only when one of the components is forced into a non-minimal bridge number conformation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends the 1992 notion of a good measure of knot complexity to a good measure F of spanning-link complexity for k-component links, using tangle products to establish monotonicity under prime-factor insertion. It then proves that all but exponentially few size-m 2SAPs have F growing at least linearly in m as m→∞, building on Atapour et al.'s pattern-counting argument for non-split links. Additional results characterize which links embed as 2SAPs in a given tube and identify geometric restrictions (e.g., forced non-minimal bridge number).

Significance. If the linear-growth theorem is rigorous, the work supplies a quantitative, asymptotically linear measure of entanglement complexity for confined polymer models, strengthening the 2010 linking-probability result. The tangle-product framework and verification that classical invariants (bridge number, splitting number) are good measures provide reusable tools for link complexity under confinement.

major comments (2)
  1. [Tangle-product extension and main asymptotic theorem] The extension of good measures via tangle products (the step that converts knot-complexity monotonicity into link-complexity additivity) does not explicitly verify that multiple inserted patterns remain additively independent once embedded under the global spanning and self-avoidance constraints of the tube; geometric interactions such as shared arcs could produce sublinear total F, directly undermining the Θ(m) lower bound obtained from the pattern-counting argument.
  2. [Main theorem on linear growth of F] When the paper invokes Atapour et al. to guarantee Θ(m) pattern insertions with high probability, it assumes the chosen invariant (e.g., bridge number) remains strictly additive after each insertion inside the tube; no separate argument rules out forced cancellations or minimal-conformation constraints that would cap the cumulative contribution below linear.
minor comments (2)
  1. [Abstract] The abstract uses 'eg' instead of 'e.g.' and should list the specific invariants shown to be good measures rather than giving examples in passing.
  2. [Definitions section] Notation for the spanning-link measure F is introduced without an explicit comparison table to the original 1992 knot measure; a short table would clarify which axioms are preserved.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and insightful comments, which help clarify the presentation of our tangle-product framework and the asymptotic result. We respond point by point to the major comments below, indicating revisions that will strengthen the manuscript.

read point-by-point responses

  1. Referee: [Tangle-product extension and main asymptotic theorem] The extension of good measures via tangle products (the step that converts knot-complexity monotonicity into link-complexity additivity) does not explicitly verify that multiple inserted patterns remain additively independent once embedded under the global spanning and self-avoidance constraints of the tube; geometric interactions such as shared arcs could produce sublinear total F, directly undermining the Θ(m) lower bound obtained from the pattern-counting argument.

    Authors: The tangle-product construction concatenates patterns sequentially along the tube axis in disjoint lattice segments, so that the monotonicity property of F applies independently to each insertion. Self-avoidance and the fixed cross-section of the tube prevent arc-sharing interactions that would reduce the total measure. We acknowledge that an explicit independence argument would improve clarity. We will add a short lemma in the revised manuscript proving that multiple disjoint pattern insertions yield strictly additive contributions to F for any good measure under the 2SAP embedding constraints. revision: yes

  2. Referee: [Main theorem on linear growth of F] When the paper invokes Atapour et al. to guarantee Θ(m) pattern insertions with high probability, it assumes the chosen invariant (e.g., bridge number) remains strictly additive after each insertion inside the tube; no separate argument rules out forced cancellations or minimal-conformation constraints that would cap the cumulative contribution below linear.

    Authors: By definition, a good measure F increases monotonically with each prime-factor insertion via the tangle product and admits no cancellations. Atapour et al. supply the high-probability count of Θ(m) embeddable non-trivial patterns; our extension then directly yields the linear lower bound on F. Tube geometry is already incorporated in the pattern-selection step, which excludes conformations that would force a reduction in the invariant. We agree a brief clarifying remark would be helpful and will insert one (or a short proof for bridge number and splitting number) in the revision. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained; no reduction to inputs by construction

full rationale

The paper defines a new extension F of good measures to spanning links via tangle-product constructions, proves monotonicity under prime insertions as a fresh result, and combines it with the external 2010 pattern-counting argument to obtain the linear lower bound on F for most 2SAPs. No equation or step equates the target asymptotic to a fitted parameter, self-definition, or unverified self-citation chain; the central claim rests on independent monotonicity and probability arguments that do not presuppose the linear growth.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The paper introduces the F measure and relies on standard knot theory and prior lattice polygon results without introducing new physical entities or fitted parameters.

axioms (3)
  • standard math The 1992 definition of good measures of knot complexity
    Extended to links in this paper.
  • standard math Properties of tangle products in knot theory
    Used to relate component knot complexity to link complexity.
  • domain assumption Results from Atapour et al. (2010) on non-split 2SAPs
    Used as base for the exponential rarity of split links.
invented entities (1)
  • Good measure F of spanning-link complexity no independent evidence
    purpose: Quantify entanglement complexity of 2SAPs extending knot measures
    Newly defined in the paper based on tangle products.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A central limit theorem for the signatures of 2-bridge knots

    math.GT 2026-04 conditional novelty 6.0

    A closed formula for the number of 2-bridge knots of crossing number c and signature σ is given, from which the signatures are shown to converge in distribution to a normal law as c tends to infinity.

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