arxiv: 2605.14593 · v1 · submitted 2026-05-14 · 🧮 math.GT

Recognition: no theorem link

Quandle presentations of surface knots in 4-manifolds and bridge numbers

Xiaozhou Zhou

Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:14 UTC · model grok-4.3

classification 🧮 math.GT

keywords surface knotfundamental quandle4-manifoldWirtinger presentationbridge numberbanded unlink diagramknot group

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The pith

Banded unlink diagrams give Wirtinger presentations for the fundamental quandle of any surface link in a 4-manifold.

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

The paper constructs explicit generators and relations for the fundamental quandle of a surface link embedded in an arbitrary 4-manifold. Earlier methods worked only when the ambient space was the 4-sphere. The construction starts from a banded unlink diagram and produces a Wirtinger-type presentation whose relations come from the crossings and the attaching bands. With this presentation in hand, the authors produce infinite families of surface knots that share the same bridge number yet are pairwise non-isotopic, and they separate knots whose fundamental groups are isomorphic but whose quandles differ.

Core claim

The fundamental quandle of a surface link in an arbitrary 4-manifold admits a Wirtinger-type presentation read directly from any banded unlink diagram of the link.

What carries the argument

Banded unlink diagram, from which generators are assigned to the arcs and relations are imposed at crossings and band attachments to define the fundamental quandle.

If this is right

For every b ≥ 4 and every m ≥ 0 there exist infinitely many pairwise non-local surface knots of bridge number b in CP^{2} # m CP^{2}-bar.
Infinite families of surface knots with isomorphic knot groups become distinguishable by their fundamental quandles.
Quandle invariants are now computable for surface links in any 4-manifold that admits a banded unlink diagram.

Where Pith is reading between the lines

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

The same diagrams may be used to compute other quandle-derived invariants in manifolds beyond the 4-sphere.
Bridge-number phenomena observed in the 4-sphere are expected to persist in connected sums of projective planes.
Explicit presentations open the possibility of finding minimal-bridge representatives by exhaustive search over diagrams.

Load-bearing premise

Banded unlink diagrams capture all relations of the fundamental quandle without extra constraints coming from the topology of the ambient 4-manifold.

What would settle it

A concrete surface link in a 4-manifold whose topologically defined fundamental quandle fails to obey the relations read from one of its banded unlink diagrams.

Figures

Figures reproduced from arXiv: 2605.14593 by Xiaozhou Zhou.

**Figure 1.** Figure 1: Band moves Theorem 3.2 ([HKM20]). Let X be a 4-manifold with Kirby diagram K. Suppose that S and S ′ are embedded surfaces in X, with banded unlink diagrams D = (K, L, v) and D′ = (K′ , L′ , v′ ), respectively. Then S and S ′ are isotopic if and only if D and D′ are related by a finite sequence of band moves, as described in [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 2.** Figure 2: The left-hand figure depicts a banded unlink diagram of the connected sum of the spun trefoil and CP 1 in CP 2 . By converting the unlink components into dotted circles and the bands into framed links, we obtain a Kirby diagram for the exterior of this surface in CP 2 , as illustrated in the right-hand figure. 3.2 A Wirtinger type presentation of the surface knot group For a surface link S in a 4-manifold … view at source ↗

**Figure 3.** Figure 3: 1-handle crossing relation Proposition 3.4. Let K be a Kirby diagram representing a connected 4-manifold X. Let 8 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Label the generators corresponding to the dotted link by x1, . . . , xd. Each 2-handle associated with a band induces the relation xix −1 i+1 = 1, which implies xi = xi+1 for all i. Let us denote this common generator by x. The (+1)-framed 2-handle imposes the relation x1x2 . . . xd = 1, which simplifies to x d = 1. Thus, the fundamental group of E(Cd) admits the presentation π1(E(Cd)) ∼= hx | x d = 1i, wh… view at source ↗

**Figure 5.** Figure 5: Primary relations At each crossing where the under arcs belongs to L1, label the under-strand by ai and ai+1. When the over arc is xj in x or aj in a, such crossings give rise to operator relations of the form ai+1 ≡ x¯jaixj or ai+1 ≡ a¯jaiaj , respectively. When the overstrand is a band in v or an arc in L2, it does not change the associated element, and the relation reduces to ai+1 = ai . We denote the … view at source ↗

**Figure 6.** Figure 6: Operator relations. • Additional primary relations are given by the bands bi in v. Each band bi gives rise to a primary relation x w j = xk between the two arcs xj and xk connected by bi , where w is a word obtained by reading the labels encountered along bi from xj to xk. If bi does not pass under any arcs of x or a, we declare that w ≡ 1. We denote these relations by b1, . . . , bs as well [PITH_FULL_IM… view at source ↗

**Figure 7.** Figure 7: An example of a band relation. Reading the word along the band from x1 to x5 yields the primary relation x5 = x x2ax¯4 1 . . • Additional operator relations s1, . . . , sl are obtained from the 2-handle attaching link L2. For each component of L2, we read the word in x∪a formed by the arcs passing over that component, traversing it once in the positive direction. Setting this word equal to the identity yie… view at source ↗

**Figure 8.** Figure 8: An example of a 2-handle relation. Reading the arcs passing over the 2-handle attaching link yields the operator relation x¯3x2x¯1a ≡ 1. Definition 3.7. The diagrammatic quandle Q(D) associated with D is defined as a quotient of the extended free quandle F Q(SP , SO). Specifically, Q(D) is given by the presentation Q(D) = [SP , SO | RP , RO] = SP × F(SP ∪ SO)/ ∼, where the sets of generators and relation… view at source ↗

**Figure 9.** Figure 9: Banded unlink diagram of Σ2 Lemma 4.7. Let S be a surface knot in S 4 , and let K be a finite kei. Then ColK(S#Σd) =    ColK(S) d is even {constant coloring} d is odd Proof. Since taking connected sum of degree-d algebraic curves and its mirroring does not change the relation that gives from the connected sum, K(S#Σd) and K(S#Cd) are isomorphic for every d. By Corollary 3.19, when d is even, K(S#Σd) i… view at source ↗

**Figure 10.** Figure 10: Banded unlink diagrams of P (left) and T (right) Comparing the two results: # ColRq (τ 0 2 (#l t2,q)#P) > # ColRq (τ 0 2 (#l t2,q′)#P). # ColRq (τ 0 2 (#l t2,q)#T) > # ColRq (τ 0 2 (#l t2,q′)#T). Thus, we conclude: τ 0 2 (#l t2,q)#P ≇ τ 0 2 (#l t2,q′)#P, τ 0 2 (#l t2,q)#T ≇ τ 0 2 (#l t2,q′)#T. Therefore, by varying q, we obtain infinitely many pairwise non-isotopic surfaces with the same bridge number. 4.… view at source ↗

read the original abstract

The fundamental quandle is an invariant for distinguishing surface knots, yet computable presentations have traditionally been limited to surfaces embedded in the $4$-sphere. Building on the framework of banded unlink diagrams introduced by Hughes, Kim, and Miller, we give a Wirtinger type presentation of the fundamental quandle of surface links in arbitrary $4$-manifolds. As applications, we extend the work of Sato and Tanaka to show that for any $b \geq 4$ and $m \geq 0$, there exist infinitely many pairwise non-local surface knots with bridge number $b$ in $\mathbb{C}P^2 \#m\overline{\mathbb{C}P^2}$, and we distinguish infinite families of surface knots with isomorphic knot groups, extending results of Tanaka and Taniguchi.

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.

Desk Editor's Note private letter to a colleague

This paper gives a Wirtinger-type quandle presentation for surface links in any closed 4-manifold and uses it for new infinite families with fixed bridge numbers in CP2 sums.

read the letter

The main takeaway is that this paper extends the banded unlink diagram method to produce a Wirtinger presentation for the fundamental quandle of surface links in arbitrary closed 4-manifolds, then applies it to get infinite families of non-local surface knots with bridge number b at least 4 in CP2 # m CPbar2, plus distinctions among some knots that share the same knot group. It builds directly on Hughes-Kim-Miller and pushes past the S4 restriction in Sato-Tanaka and Tanaka-Taniguchi. The applications look concrete and give explicit new examples that were not available before. The construction appears to follow the prior diagram framework closely enough to make the extension workable. The soft spot is the general case for non-simply-connected manifolds. The stress-test note correctly flags that the complement group is an extension of pi1(M), so the quandle might pick up extra relations not visible in the diagram alone. Their examples stay inside simply-connected spaces, so those results probably hold, but the general claim needs to show explicitly how the manifold topology is absorbed without missing relations. This is for people working on 4D knot invariants and quandle presentations. A reader who cares about surface knots or computable distinctions in 4-manifolds will find the new families useful. It has enough new content and concrete applications to deserve a serious referee. I recommend sending it out for peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript claims to extend the banded unlink diagrams of Hughes-Kim-Miller to obtain a Wirtinger-type presentation for the fundamental quandle of surface links in arbitrary closed 4-manifolds. It applies the construction to produce, for any b ≥ 4 and m ≥ 0, infinitely many pairwise non-local surface knots of bridge number b in CP² # m CPbar², and to distinguish infinite families of surface knots that have isomorphic knot groups.

Significance. If the presentation is rigorously established, the work supplies the first general computational tool for quandle invariants of surface links outside S⁴. The concrete applications to bridge numbers and to knots with isomorphic groups but non-isomorphic quandles furnish new families of examples that were previously unavailable, thereby strengthening the role of quandle theory in 4-dimensional knot theory.

major comments (1)

[Main theorem and construction section] The central construction (presumably the statement and proof of the Wirtinger presentation in the section following the introduction) asserts that the generators and relations coming from arcs, crossings, and bands in a banded unlink diagram already encode the full quandle structure on the complement in an arbitrary 4-manifold M. This claim is load-bearing for the general statement, yet the derivation does not isolate or verify the contribution of π₁(M) to the quandle operation when M is not simply connected; the applications are confined to the simply-connected manifolds CP² # m CPbar², leaving the general case untested.

minor comments (2)

[Abstract] The abstract and introduction would benefit from an explicit reference to the theorem number that states the Wirtinger presentation, rather than the informal phrase “we give a Wirtinger type presentation.”
[Figures and diagrams] Figure captions and diagram descriptions should indicate which relations arise from the manifold’s topology versus the link diagram itself, to help readers track the general versus special cases.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough reading and for identifying a point that requires clarification in the presentation of the main theorem. We address the concern directly below and outline the revisions we will make.

read point-by-point responses

Referee: [Main theorem and construction section] The central construction (presumably the statement and proof of the Wirtinger presentation in the section following the introduction) asserts that the generators and relations coming from arcs, crossings, and bands in a banded unlink diagram already encode the full quandle structure on the complement in an arbitrary 4-manifold M. This claim is load-bearing for the general statement, yet the derivation does not isolate or verify the contribution of π₁(M) to the quandle operation when M is not simply connected; the applications are confined to the simply-connected manifolds CP² # m CPbar², leaving the general case untested.

Authors: We agree that the current write-up does not isolate the contribution of π₁(M) in a dedicated step, which makes the general claim harder to verify. The construction proceeds by lifting the banded unlink diagram to the universal cover of M and imposing the Wirtinger relations on the lifted arcs; the deck transformations of π₁(M) act by conjugation on the quandle generators, and this action is encoded implicitly through the relations coming from the 1-handles and the paths that connect the bands. However, we did not extract this action into a separate lemma or diagram chase that explicitly tracks how an element of π₁(M) modifies a quandle operation. The applications are restricted to simply-connected manifolds precisely because the resulting quandle is then free of this extra action, allowing direct comparison with known examples. We will revise the proof section by adding a short subsection that isolates the π₁(M)-action (via a commutative diagram relating the quandle of the cover to the quandle of M) and verifies that the presentation remains complete when π₁(M) is nontrivial. This will make the general statement fully rigorous while leaving the concrete applications unchanged. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the quandle presentation derivation

full rationale

The paper constructs a Wirtinger-type presentation for the fundamental quandle of surface links in arbitrary 4-manifolds by extending the banded unlink diagrams of Hughes, Kim, and Miller. This is a direct topological construction rather than any reduction of a claimed result to a fitted parameter, self-definition, or self-citation chain. No equations or steps rename a known result as new unification, smuggle an ansatz via citation, or import uniqueness from the authors' prior work. Citations to Sato-Tanaka and Tanaka-Taniguchi supply context for applications but do not bear the load of the central claim. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard domain assumptions from quandle theory and prior diagrammatic frameworks without introducing new free parameters or invented entities.

axioms (2)

domain assumption The fundamental quandle is an invariant for surface knots
Standard assumption in quandle knot theory as referenced in the abstract.
domain assumption Banded unlink diagrams represent surface links in 4-manifolds
Framework introduced by Hughes, Kim, and Miller and extended here.

pith-pipeline@v0.9.0 · 5424 in / 1130 out tokens · 57108 ms · 2026-05-15T01:14:04.818742+00:00 · methodology

discussion (0)

Reference graph

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