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arxiv: 2410.01921 · v2 · submitted 2024-10-02 · 🧮 math.GT

Bridge Multisections of Knotted Surfaces in S⁴

Rom\'an Aranda , Carolyn Engelhardt This is my paper

Pith reviewed 2026-05-23 19:58 UTC · model grok-4.3

classification 🧮 math.GT
keywords bridge multisectionsknotted surfacesband surgerymultiplane diagramssurface links in S^4unknotting moves
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The pith

Band surgery supplies a complete set of moves connecting any two multiplane diagrams of the same knotted surface in four-space.

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

Bridge multisections give combinatorial descriptions of surface links in four-space by means of tuples of trivial tangles. The paper develops band surgery as an operation on the associated multiplane diagrams. It proves that this operation, together with other moves, relates any two diagrams that represent the same surface up to isotopy. A reader would care because the result turns the diagrams into a practical tool for working with these surfaces rather than abstract objects.

Core claim

The authors prove a uniqueness result for bridge multisections: any two multiplane diagrams of the same surface are related by a sequence of band surgeries. Band surgery is a surgery operation on multiplane diagrams. As applications, any n-valent graph with an n-edge coloring is the spine of a bridge multisection for an unknotted surface, and any multisected surface in S^4 can be unknotted by finitely many band surgeries.

What carries the argument

Band surgery, the surgery operation on multiplane diagrams that changes the diagram while keeping the isotopy class of the surface fixed.

If this is right

  • Any two multiplane diagrams of the same surface are related by band surgeries.
  • Any n-valent graph with an n-edge coloring arises as the spine of a bridge multisection of an unknotted surface.
  • Every multisected surface in S^4 can be reduced to an unknotted one by a finite sequence of band surgeries.

Where Pith is reading between the lines

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

  • Diagrams of the same surface can be transformed into each other, which may support the development of invariants or recognition algorithms based on diagram simplification.
  • Bridge multisections may extend to other 4-manifolds or higher-dimensional links through similar diagrammatic techniques.

Load-bearing premise

Band surgery preserves the isotopy class of the embedded surface while only changing its diagram.

What would settle it

Two multiplane diagrams representing isotopic surfaces that cannot be connected by any sequence of the given moves including band surgery.

Figures

Figures reproduced from arXiv: 2410.01921 by Carolyn Engelhardt, Rom\'an Aranda.

Figure 1
Figure 1. Figure 1: Turning a (1; 1)–bridge 4–section of an unknotted 2–sphere into a (2; 1)–bridge 4–section of a Klein bottle. tube [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Turning a (2; 2, 2, 2, 2)–bridge quadrisection of a 2–component unlink of 2–spheres into a (3; 2, 2, 2)–bridge trisection of a spun trefoil. In Section 6, we prove any 1–handle addition on a multisected surface can be achieved by a sequence of band surgeries. Thus, band surgery is an unknotting move for multiplane diagrams, just as 1–handle addition can unknot any surface in 4–space [HK79]. Theorem 1.2. Le… view at source ↗
Figure 3
Figure 3. Figure 3: Standard projective planes determined by different choices of crossing. Given an embedded surface F ⊂ S 4 , the crosscap sum of F is the new surface obtained by adding a P± summand to F. We say that F ′ is obtained from F by 1–handle addition if F ′ is the result of replacing two small disks in F with a thin tube connecting them [CKS04b]. More precisely, if D2 × I is an embedded 1–handle for F, so that F ∩… view at source ↗
Figure 4
Figure 4. Figure 4: Three kinds of bands. Remark 2.3 (Bands as framed arcs). In Section 4.2, it will be useful to describe a band as in case (b) or (c) using a pair (α, f) where α is an embedded arc with framing f. Suppose T and T ′ are trivial tangles with the same endpoints. If (α, f) represents a band for a properly embedded 1–manifold L, we denote by L[α, f] the result of L after banded surgery. If (α, f) and (α ′ , f′ ) … view at source ↗
Figure 5
Figure 5. Figure 5: An example of dual bands for a crossingless trivial tangle. Lemma 2.4 (Lem 3.3 of [MZ17]). Let T1 and T2 be two b–bridge trivial tangles with the same endpoints. If T1 ∪ T2 is a c–component unlink, then there exist a set v of (b − c) bands such that v is dual to T1, and T1[v] = T2. 3. Bridge multisections We discuss a 4-dimensional analog of bridge position for links in S 3 . Bridge multisections can be de… view at source ↗
Figure 6
Figure 6. Figure 6: In [ABG+23, Sec 4.3], the authors explained how (Ta(β), Tb(β, m), Tc(β), Tb(β, 0)) is a multiplane diagram for a (2b − 1, b)–bridge 4–section of the m–twist spun of K(β). In fact, (Ta(β), Tb(β, m), Tc(β)) is a bisection for the ribbon disk obtained by m–twisting the half spin of K(β). The following is a consequence of the observation that the tuple (Tc(β), Tb(β, m), Ta(β)) represents the same ribbon disk i… view at source ↗
Figure 7
Figure 7. Figure 7: Model for arcs ρi and ρi+1 satisfying conditions (3) and (4). Lemma 4.2. Let ρ be a collection of admissible arcs for T . For every i = 1, . . . n, there is an embedded disk ∆i ⊂ Xi with interior in int(Xi) satisfying the following conditions: (1) ∂∆i is the point union of ∂∆i ∩ ∂Xi = ρi ∪ ρi+1 and an arc δi properly embedded in Xi. (a) If both ρi and ρi+i are type 1 and ρi∪ρi+1 connects the same component… view at source ↗
Figure 8
Figure 8. Figure 8: Two kinds of framings satisfy that Li [ρi ∪ ρi+1, f] is an unlink. Definition 4.4. Let (ρ, f) = {(ρi , fi)} n i=1 be a tuple of framed arcs in a multiplane diagram T . We say that (ρ, f) is admissible if ρ is a collection of admissible arcs for T , the framings {fi} n i=1 agree at ∗ ∈ int(Σ), and the framing fi + fi+1 is zero or ±1 whenever ρi ∪ ρi+1 connects the same component of Li . Note that framing on… view at source ↗
Figure 9
Figure 9. Figure 9: How the disks Di change after surgery along an admissible tuple (ρ, f). Example 4.6 (Tubings). Fix 1 ≤ m ≤ n − 2 and consider an admissible tuple (ρ, f) having the following properties: (1) the arcs ρi are type 1 for i = 1, . . . , m + 1 and type 0 otherwise, and (2) for i = 1, . . . , m, the framing fi + fi+1 is zero whenever ρi ∪ ρi+1 connects the same component of Li . If r is the number of arcs ρi ∪ ρi… view at source ↗
Figure 10
Figure 10. Figure 10: Turning a (3; 2, 2, 3)–bridge trisection of a 2–component unlink of 2–spheres into a (4; 2)–bridge trisection of a spun knot. The top surface is unknotted by Proposition 3.9. Example 4.7 (More Tubings). Consider an admissible tuple (ρ, f) satisfying the following condition: (1) if ρi and ρj type 1 arcs for some i 6= j, then 2 < |i − j| < n − 1. If t is the number of type 1 arcs, then the surface FT [ρ,f] … view at source ↗
Figure 11
Figure 11. Figure 11: Different choices of arcs yield surfaces related by 1–handle addition. Remark 4.10 (Counting 1–handles and crosscaps). Deciding how many 1–handles and crosscaps are added between FT and FT [ρ,f] is a combinatorial process. Suppose we are given an admissible tuple (ρ, f) with ρ1 a type 0 arc. Then write a string in the characters {0, 1, 0=, ±=, 6=} by the following rules: (1) First write the types of the ρ… view at source ↗
Figure 12
Figure 12. Figure 12: Different choices of arcs yield surfaces related by at most one crosscap sum. surgery 1-handle addition surgery [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Different choices of arcs yield surfaces related by 1–handle addition. fi + fi+1 is zero or ±1. If ρi ∪ ρi+1 connects distinct components of Li , add a 6= symbol in between. For example, the string we obtain from the tuple in [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Turning a (2; 1)–bridge 4–section into a (3; 1)–trisection of an unknotted torus. perturb merge [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Turning a (2; 1)–bridge 4–section into a (3; 1)–trisection of a Klein bottle. Proposition 5.7. Let n ≥ 4 and let T = (T1, T2, T3, . . . , Tn) be a (b; c1, c2, . . . , cn)–multiplane diagram. There exist (b − c2) two-sector perturbations such that the resulting multiplane diagram (T ′ 1 , T ′ 2 , T ′ 3 , . . . , T ′ n ) can be merged into a multisection with (n − 1) sectors (T ′ 1 , T ′ 3 , T ′ 4 , . . . ,… view at source ↗
Figure 16
Figure 16. Figure 16: Two–sector perturbations. We observe in [PITH_FULL_IMAGE:figures/full_fig_p017_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The links T ′ 1 ∪ T ′ 3 and T1 ∪ T2 are isotopic. (a) (b) (c) [PITH_FULL_IMAGE:figures/full_fig_p017_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: From (a) to (b), one slides the purple punctures to the left. The resulting triplane diagram admits (b − c2) deperturbations turning (b) into (c). We now explain the perturbation-merging process in Proposition 5.7 using bands instead of standard pairs. This may be useful in practice as dual bands can guide the necessary 2–sector perturbations [PITH_FULL_IMAGE:figures/full_fig_p017_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Turning a (3; 2)–bridge 4–section into a (4; 2)–trisection of an m–twist spun knot. perturb [PITH_FULL_IMAGE:figures/full_fig_p018_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Local modification around the endpoint of a band in v2. Proof. Let T1 and T2 be two multiplane diagrams with isotopic underlying surfaces. Proposition 5.7 applied to each Ti gives a sequence of two-sector perturbations and merging moves turning Ti into a triplane diagram T ′ i . To end, by [MZ17, Thm 1.7], there is a finite sequence of triplane moves (thus multiplane moves) relating T ′ 1 with T ′ 2 . 6. … view at source ↗
Figure 21
Figure 21. Figure 21: Top row is a guiding arc t that intersects a triplane diagram with conditions (a) and (b). From top to middle, we dragged the minimum of the blue arc in T1 towards Σ1. Thus, getting red/blue tangles with only maxima satisfying (c). From middle to bottom, we perform a 0–sector perturbation using as the type 1 arc the blue arc connecting the point c with the tangle T3. By transversality, t can be isotoped t… view at source ↗
Figure 22
Figure 22. Figure 22: Sequence of perturbations to make t = ρ1 ∪ ρ2 an admissible pair of arcs. We are ready to prove the main results of this section. This relies heavily on [HK79, Thm 2.3] which states that any surface in 4-space can be unknotted by adding enough 1–handles. In particular, any pair of surfaces can be related by a sequence of 1–handle additions, connected sums with unknotted projective spaces, and their invers… view at source ↗
Figure 23
Figure 23. Figure 23: Two edge-compressions turning the graph K6 into a graph with two vertices [PITH_FULL_IMAGE:figures/full_fig_p020_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: How to isotope Li (right) instead of performing a destabilization (left). At the end of the above process, we are left with a b–bridge crossingless diagram for Li and a type 1 arc ρi that braids with some strands of Li . We can then choose ρi+1 so that it connects {∗} = ρi ∩ int(Σ) with our strand ti+1 while undoing all the braiding caused by ρi . This way, ρi ∪ ρi+1 can be isotoped (via Reidemeister II m… view at source ↗
Figure 25
Figure 25. Figure 25: Guided by the compressions in [PITH_FULL_IMAGE:figures/full_fig_p022_25.png] view at source ↗
read the original abstract

Bridge multisections are combinatorial descriptions of surface links in 4-space using tuples of trivial tangles. They were introduced by Islambouli, Karimi, Lambert-Cole, and Meier to study curves in rational surfaces. In this paper, we prove a uniqueness result for bridge multisections of surfaces in 4-space: we give a complete set of moves relating to any two multiplane diagrams of the same surface. This is done by developing a surgery operation on multiplane diagrams called band surgery. Another application of this surgery move is that any $n$-valent graph with an $n$-edge coloring is the spine of a bridge multisection for an unknotted surface. We also prove that any multisected surface in $S^4$ can be unknotted by finitely many band surgeries.

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 introduces bridge multisections of knotted surfaces in S^4 via tuples of trivial tangles and multiplane diagrams. It claims to prove a uniqueness theorem by exhibiting a complete set of moves between diagrams of the same surface, achieved through a new operation called band surgery on multiplane diagrams. Additional results state that any n-valent graph equipped with an n-edge coloring arises as the spine of a bridge multisection of an unknotted surface, and that any multisected surface in S^4 can be unknotted via finitely many band surgeries.

Significance. If the invariance of band surgery under isotopy and the completeness of the resulting moves are established, the work supplies a combinatorial calculus for surface links in 4-space that extends earlier constructions of bridge multisections. The unknitting theorem and the realization result for colored graphs would then constitute concrete applications with potential utility for unknotting questions and diagram-based invariants in 4-dimensional topology.

major comments (2)
  1. [Section defining band surgery] The central uniqueness claim rests on the assertion that band surgery preserves the isotopy class of the embedded surface while only changing the diagram. This step is load-bearing for both the completeness of the moves and the unknitting result; a detailed verification (including how the surgery is performed on the underlying 4-manifold and why the resulting surface remains isotopic) is required in the section introducing the operation.
  2. [Section on realization of colored graphs] The statement that any n-valent n-edge-colored graph is the spine of a bridge multisection of an unknotted surface is derived from the surgery move; the construction must be shown to produce a surface whose bridge multisection recovers the given colored graph, with explicit checks that the resulting surface is unknotted.
minor comments (2)
  1. [Introduction and definitions] Notation for multiplane diagrams and the precise relation between bridge multisections and the diagrams should be clarified early, with a table or diagram summarizing the correspondence between combinatorial data and the embedded surface.
  2. [Uniqueness theorem] The abstract states that proofs exist; the manuscript should include explicit references to the relevant lemmas or propositions when invoking the preservation property in later arguments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Section defining band surgery] The central uniqueness claim rests on the assertion that band surgery preserves the isotopy class of the embedded surface while only changing the diagram. This step is load-bearing for both the completeness of the moves and the unknitting result; a detailed verification (including how the surgery is performed on the underlying 4-manifold and why the resulting surface remains isotopic) is required in the section introducing the operation.

    Authors: We agree that the current exposition would benefit from greater explicitness. In the revised manuscript we will expand the definition of band surgery with a self-contained verification: we will describe the operation directly on the underlying 4-manifold (specifying the 4-dimensional handle attachments and the resulting change to the surface), and we will prove that the new surface is isotopic to the original by exhibiting an explicit isotopy supported in a neighborhood of the surgery band. This added material will make the invariance statement fully rigorous and will serve as the foundation for both the move-completeness theorem and the unknitting result. revision: yes

  2. Referee: [Section on realization of colored graphs] The statement that any n-valent n-edge-colored graph is the spine of a bridge multisection of an unknotted surface is derived from the surgery move; the construction must be shown to produce a surface whose bridge multisection recovers the given colored graph, with explicit checks that the resulting surface is unknotted.

    Authors: We accept that the realization argument requires more explicit verification. In the revision we will rewrite the construction to proceed in two clear steps: first, we apply a sequence of band surgeries to a standard unknotted surface whose multiplane diagram is known, producing a diagram whose spine is the prescribed colored graph; second, we verify directly that the resulting surface is unknotted by exhibiting a sequence of isotopies (or equivalently, by showing that its fundamental group and Seifert form match those of the unknot). These checks will be inserted immediately after the construction so that the claim is fully substantiated. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is self-contained via explicit definitions and standard topological arguments

full rationale

The paper defines bridge multisections and the new band surgery operation directly on multiplane diagrams, then proves (within the manuscript) that the operation preserves isotopy class of the embedded surface while relating diagrams. The completeness of the moves, the unknitting result, and the spine realization for colored graphs all follow from these definitions plus standard 4-dimensional isotopy and handlebody arguments. No equations reduce by construction to fitted inputs, no load-bearing self-citations appear, and the cited prior work (Islambouli et al.) is external and non-overlapping with the present authors. The derivation chain is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard definitions of tangles, multiplane diagrams, and isotopy of surfaces in S^4 from the cited introduction of bridge multisections; no free parameters, invented entities, or ad-hoc axioms are introduced beyond the new band surgery move itself.

axioms (2)
  • domain assumption Multiplane diagrams faithfully encode surface links in 4-space up to isotopy
    Invoked when claiming that moves relate diagrams of the same surface.
  • domain assumption Band surgery is a well-defined local operation on diagrams that preserves the embedded surface
    Central to the uniqueness proof and unknitting application.

pith-pipeline@v0.9.0 · 5664 in / 1291 out tokens · 22186 ms · 2026-05-23T19:58:55.536503+00:00 · methodology

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Reference graph

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