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arxiv: 2606.29452 · v1 · pith:INSNGFZ5new · submitted 2026-06-28 · 🧮 math.GT · math.QA

Handle decompositions and the 1-dimensional inputs skein lasagna module

Imogen Montague , Ian A. Sullivan This is my paper

Pith reviewed 2026-06-30 01:56 UTC · model grok-4.3

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keywords skein lasagna module1-dimensional inputshandle decompositionsRozansky-Willis homologylasso relation4-manifoldsKhovanov homology
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The pith

The Khovanov skein lasagna module with 1-dimensional inputs for 4-manifolds built from 1- and 2-handles equals a cabled colimit of Rozansky-Willis homologies modulo the lasso relation.

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

The paper develops formulas that describe how the skein lasagna module changes under the attachment of 1-handles and 2-handles. These formulas reduce the computation of the module, defined over the rationals, to a cabled colimit of Rozansky-Willis homologies after imposing one additional relation called the lasso relation. Concrete values are worked out for disk bundles over the 2-sphere. A partial vanishing result is proved for the product of a surface of genus at least one with a disk.

Core claim

We establish handle attachment formulas for the Khovanov skein lasagna module with 1-dimensional inputs over Q. For a 4-manifold built out of 1- and 2-handles, the invariant can be computed in terms of a cabled colimit of Rozansky-Willis homologies, modulo a new relation which we call the lasso relation. We then present some explicit calculations for disk bundles over S^2, as well as a partial vanishing result for 4-manifolds of the form Σ_g × D^2, g ≥ 1.

What carries the argument

Handle attachment formulas that reduce the 1-dimensional inputs skein lasagna module to a cabled colimit of Rozansky-Willis homologies subject to the lasso relation.

If this is right

  • The invariant takes explicit values on disk bundles over S^2.
  • The invariant vanishes in some degrees for every 4-manifold of the form Σ_g × D^2 when g ≥ 1.
  • Any 4-manifold presented by 1- and 2-handles has its module determined by Rozansky-Willis data plus the lasso relation.

Where Pith is reading between the lines

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

  • The lasso relation may admit a direct diagrammatic description independent of the colimit construction.
  • The same reduction technique could apply to manifolds that also involve 3-handles once functoriality for those attachments is verified.
  • The resulting values might be compared with other 4-manifold invariants that are also defined via handle decompositions.

Load-bearing premise

The skein lasagna module with 1-dimensional inputs is well-defined and satisfies the expected functoriality under handle attachments.

What would settle it

An explicit 4-manifold whose skein lasagna module, computed directly, differs from the value obtained from the cabled colimit after imposing the lasso relation.

Figures

Figures reproduced from arXiv: 2606.29452 by Ian A. Sullivan, Imogen Montague.

Figure 1
Figure 1. Figure 1: A cartoon of the lasso move isotopy. constructed a gl2 -refinement of Rozansky-Willis homology and showed it satisfied the func￾toriality properties necessary to construct a skein lasagna module. One can extract from their construction two link homology theories which we denote in this paper by KhR− 2,O(L) and KhR− 2,T (L); these are defined for (framed, oriented) null-homologous and two-divisible links L … view at source ↗
Figure 2
Figure 2. Figure 2: TOP: βi movie, BOTTOM: Ei movie. Components other than the ith and (i + 1)th omitted. that permutes the parallel strands of Kn. More specifically, for each generator σi ∈ Bn, there is an associated endomorphism βi ∈ End(S 2 0 (X; Kn)) defined by pulling back the action of σi on an n-punctured D2 × S 1 to a tubular neighborhood of K ⊂ ∂X. When X = B4 , we remark that this action factors through the symmetri… view at source ↗
Figure 3
Figure 3. Figure 3: The standard and twisted belt links ̃1(k,l) and ̃1(k,l)p in S 1 ×S 2 for p ∈ Z. where we have a disjoint copy of K inside of Y ∖ B3 ⊂ Y #S 1 × S 2 , along with a disjoint copy of ̃1(1, 1) ⊂ S 1 × S 2 ∖ B3 ⊂ Y #S 1 × S 2 . Choose a framed arc a connecting the i￾th component Ki ⊂ K with the positively oriented component ̃1(1, 0) ⊂ ̃1(1, 1). We then define K(i) ⊂ Y #S 1 × S 2 to be the framed link obtained by… view at source ↗
Figure 4
Figure 4. Figure 4: Cancelling one- and two-handles. Here, K˚i denotes the (1, 1)- tangle in Y ∖ B3 obtained from (Y, Ki) by deleting a 3-ball/arc pair. denote the cobordism of pairs given by capping off the unknot component via a disk contained in B3 × [0, 1 2 ] ⊂ Y × [0, 1], and attaching trivial cylinders to remaining link components. We then define (Wi , Σi) ∶= (Y × [0, 1], C) ○ (Zi , Si), (W′ i , Σ ′ i ) ∶= (Y × [0, 1], … view at source ↗
Figure 5
Figure 5. Figure 5: Top Left: A link K = K1 ∪ K2 ⊂ #3S 1 ×S 2 . Top Right: the link K(1) ⊂ #4S 1 × S 2 . Bottom: A cable K(k ±; ℓ ±; 1) ⊂ #4S 1 × S 2 of K(1). obtained by appropriately taking framed push-offs of the surface components of the cobor￾disms (Wi , Σi) and (W′ i , Σ′ i ) defined above, with analogous decompositions (Wi , Σk±,ℓ±,i) = (Y × [0, 1], Cℓ ±) ○ (Zi , Sk±,ℓ± (4.6) ,i), (W′ i , Σ ′ k±,ℓ±,i) = (Y × [0, 1], Ck… view at source ↗
Figure 6
Figure 6. Figure 6: Handlesliding γ over Ki . Corollary 4.5. Let (X, K, L) and (X′ , L′) be as above with X = ♮ n S 1 × B3 . Then for ◇ ∈ {O, T}, the type ◇ 1-dimensional input skein lasagna module of (X′ , L′) at level α can be expressed by the formula S 2,◇ 0 (X′ ;L ′ ;α) ≅ Coeq( m ⊕ i=1 KhR− 2,◇,(α,αi) (#n+1S 1×S 2 ; K(i), L) Φα ◇ Ψα ◇ KhR− 2,◇,α(#nS 1×S 2 ; K, L)). The rest of this section is devoted to the proof of Theor… view at source ↗
Figure 7
Figure 7. Figure 7: (From left to right) The cap, cup and switch critical points in [0, 1] × Gi . isotope the surface Σ so that its intersection with the 2-handles is precisely a union of core parallel disks. By removing these 2-handles along with these core-parallel disks, we obtain a lasagna filling (f (α,η) O ) −1(F) of (X, K(k +,k −) ∪ L) for some k ± ∈ Nm satisfying k + − k − = α. In order to show that (f (α,η) O ) −1 is… view at source ↗
Figure 8
Figure 8. Figure 8: A schematic of a piece of 1-dimensional input passing through a co-core of the 2-handle Hi ≈ D2 × D2 after projecting down to the core disk D2 × {0}, as in the proof of Lemma 4.7. The co-core Gi is represented by the central dot {0} × {0}, the piece of 1-dimensional input passing through Hi is depicted by the white strip, the half disks D± L,i and D± R,i are shown in blue and red, respectively, and the dis… view at source ↗
Figure 9
Figure 9. Figure 9: Using the enclosement relation to subsume the remaining sheets into the “switch” of the switch critical point, as in the proof of Lemma 4.7 [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The decomposition of the union of ν(Ki) and a 4-dimensional 1-handle as specified by (4.14). where a, b ∈ S 2 denote the north and south poles, respectively (see [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: A depiction of ψk ±,ℓ±,1 for the 0-framed unknot U. We can therefore replace (4.15) with the equation (4.16) KhR− 2,◇ (Z ′ i , S′ k±,ℓ±,i) = ϵ ○ KhR− 2,◇ (ψk±,ℓ±,i). where ψk±,ℓ±,i is as in Proposition 4.11. Example 4.12. Let U = (U, 0) ⊂ ∂B4 be the 0-framed unknot. We can take ψk±,ℓ±,i as in Proposition 4.11 to be given by inv followed by a planar isotopy as pictured in [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
Figure 12
Figure 12. Figure 12: A depiction of ψk ±,ℓ±,1 for the p-framed unknot (U, p). The map (∗) is given by repeating steps 4–6 ∣p − 1∣ times. Depicted is the case where p ≥ 0; the case p < 0 can be obtained by changing all of the crossings in the bottom left picture, and changing the +1-twists to −1-twists and vice-versa in the first three pictures on the bottom. S 2,◇ 0 (Y × [0, 1], Cℓ ±) = id ⊗ εℓ ± ∶ S 2,◇ 0 (X, K(k + ,k − ) ∪ … view at source ↗
Figure 13
Figure 13. Figure 13: The surface P used in the proof of Proposition 4.15 [PITH_FULL_IMAGE:figures/full_fig_p035_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The cobordisms (Wi , Σk±,k± i +1,i) and (W′ i , Σ′ k±,k± i +1,i), respectively, after gluing in the lasagna filling Fu as defined in the proof of Proposition 4.15. via fu ∶= gl(− ⊗ [Fu]) where gl denotes the gluing map from Proposition 3.6. Given v ∈ S 2,◇ 0 (X; K(k +,k −) ∪ L), the elements v ′ , v′′ as in the statement of the proposition are given by f1(v), fX(v), respectively. Observe that (Φ◇,k±,k± i … view at source ↗
Figure 15
Figure 15. Figure 15: The matching δr,k± =((1 ⊗ X + X ⊗ 1) ⊗k + ⊗ X⊗(α ++r−2k +) ) ⊗ ((1 ⊗ X + X ⊗ 1) ⊗k − ⊗ X⊗(α −+r−2k −) ) ⊗ (X⊗(α ++r) ) ⊗ (X⊗(α −+r) ), so that in particular we have that v ∈ im(g ○ ϵ) = im(Fα±+r,α±+r) by the preceding discussion. Then (id ⊗ εα±+r)(v) = ((1 ⊗ X + X ⊗ 1) ⊗k + ⊗ X⊗(α ++r−2k +) ) ⊗ ((1 ⊗ X + X ⊗ 1) ⊗k − ⊗ X⊗(α −+r−2k −) ) ∼ 2 k ++k − xr,k±, (symmetry relations) whereas (εα±+r ⊗ id)(v) = 0, as… view at source ↗
Figure 16
Figure 16. Figure 16: The relevant diagram for computing Φ◇,k±,ℓ±,1 and Ψ◇,k±,ℓ±,1 for D(p). (1) The counit map ϵ ∶ KhR− 2,◇ (̃1(k + , k− )p) → KhR2,◇(T(k + + k − , p(k + + k − ))k +,k−) is an isomorphism in bi-degree (0, k+ + k −). (2) Let ε̃◇,k±,p denote the composition of maps KhR0,k++k − 2,◇ (T(k + + k − , p(k + + k − ))k +,k−) (ϵ) −1 ÐÐ→ KhR−,0,k++k − 2,◇ (̃1(k + , k− )p) τ −p ◇ ÐÐ→ KhR−,0,k++k − 2,◇ (̃1(k + , k− )) ϵÐ→ K… view at source ↗
Figure 17
Figure 17. Figure 17: The maps ΦO,r,s,1 and ΨO,r,s,1 for the p-framed longitude of S 1 × S 2 . such that the following diagram commutes: (6.7) KhR−,0,2r 2,O (S 1 × S 2 ,̃1(r, r)p) KhR−,0,2r+2 2,O (S 1 × S 2 ,̃1(r + 1, r + 1)p) Q. fO,r,p ≅ ψ [1] O ≅ fO,r+1,p ≅ (2) We have a short exact sequence (6.8) ⊕ (h,q)≠(0,2r) KhR−,h,q 2,O (S 1 × S 2 ,̃1(r, r)p) iÐ→ KhR− 2,O(S 1 × S 2 ,̃1(r, r)p) ε ′ O,r,p ÐÐÐ→ Q where i denotes the canoni… view at source ↗
Figure 18
Figure 18. Figure 18: A diffeomorphism of pairs. The map (∗) is homotopy equivalence given by pulling the p-full twist FT⊗p 2n through a Rozansky projector [PITH_FULL_IMAGE:figures/full_fig_p049_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: A sequence of isomorphisms interpolating between Lr,s,p and ̃1(r, r) ⊔ ̃1(s, s). Proof. By naturality of the Gluck twist isomorphism τ p O , it suffices to prove the result for p = 0. For (1), note that we have an isomorphism (6.9) KhR−,0,2r 2,O (S 1 × S 2 ,̃1(r, r)) hÐ→≅ HH0(Hr ⊗ Q)2r, where Hr denotes Khovanov’s arc algebra as defined in [Kho02], and HH0(Hr⊗Q)2r denotes the piece of the zeroth Hochschil… view at source ↗
Figure 20
Figure 20. Figure 20: Alternate maps Φ′ O,r,s,1 and Ψ′ O,r,s,1 . We then set fO,r,0 to be the composition of these isomorphisms. Furthermore, h intertwines the dotted annulus homomorphism ψ [1] O ∶ KhR−,0,2r 2,O (S 1 × S 2 ,̃1(r, r)) → KhR−,0,2r+2 2,O (S 1 × S 2 ,̃1(r + 1, r + 1)) with the map ϕ ∶ HH0(Hr ⊗ Q)2r ≅Ð→ HH0(Hr+1 ⊗ Q)2r+2 xr ↦ xr+1, which implies commutativity of (6.7). For (2), it suffices to show that: (a) The map… view at source ↗
Figure 21
Figure 21. Figure 21: Left: A Kirby diagram of B4 . Right: A Kirby diagram for Σg × D2 , the disk bundle over the genus g surface Σg with Euler number 0, built from 2g 1-handles and a single 0-framed 2-handle. Proof. There exists a Kirby diagram of Σg × D2 consisting of a single 0-framed crossingless unknot linked with 2g dotted 1-handles given by a crossingless unlink such that the diagram of the corresponding knot K ⊂ #2gS 1… view at source ↗
read the original abstract

We establish handle attachment formulas for the Khovanov skein lasagna module with 1-dimensional inputs over $\mathbb{Q}$, defined recently by Ren, Wedrich, Willis, Zhang, and the second author. For a $4$-manifold built out of $1$- and $2$-handles, the invariant can be computed in terms of a cabled colimit of Rozansky-Willis homologies, modulo a new relation which we call the lasso relation. We then present some explicit calculations for disk bundles over $S^{2}$, as well as a partial vanishing result for $4$-manifolds of the form $\Sigma_{g}\times D^{2}$, $g\geq 1$.

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

0 major / 3 minor

Summary. The manuscript establishes handle attachment formulas for the Khovanov skein lasagna module with 1-dimensional inputs over ℚ. For a 4-manifold built from 1- and 2-handles, the invariant is expressed as a cabled colimit of Rozansky-Willis homologies modulo a new lasso relation. Explicit calculations are given for disk bundles over S² together with a partial vanishing result for manifolds of the form Σ_g × D² (g ≥ 1).

Significance. If the formulas hold, the work supplies a concrete computational bridge between the 1-dimensional-input skein lasagna module and the more established Rozansky-Willis homology, together with immediate concrete output in the form of the disk-bundle calculations and the vanishing statement. These are genuine strengths that increase the utility of the invariant.

minor comments (3)
  1. The lasso relation is introduced as new; a short self-contained definition or diagrammatic presentation in the main text (rather than only in the colimit statement) would improve readability.
  2. Notation for the cabled colimit should be introduced once with a clear reference to the cabling parameter and the precise colimit category.
  3. The partial vanishing result for Σ_g × D² would benefit from an explicit statement of the range of g for which vanishing is proved versus conjectured.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance in bridging the 1-dimensional-input skein lasagna module to Rozansky-Willis homology, and recommendation for minor revision. The referee's description accurately captures the handle attachment formulas, the lasso relation, the disk-bundle calculations, and the partial vanishing result.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained given prior definition

full rationale

The paper takes the 1-dimensional-input skein lasagna module as defined in prior cited work (including the second author) and derives new handle-attachment formulas expressing it via cabled colimits of Rozansky-Willis homologies modulo the lasso relation. No equation or step in the abstract or stated claims reduces a derived quantity to a fitted input or self-citation by construction; the functoriality assumption is explicitly external, and the new computations (explicit disk-bundle calculations, partial vanishing) add independent content. This matches the default expectation of non-circularity for papers that build on established prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The work relies on the prior definition of the skein lasagna module and on standard properties of handle decompositions in 4-manifold topology.

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Reference graph

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