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arxiv: 2606.30823 · v1 · pith:2YSVBMMPnew · submitted 2026-06-29 · 🧮 math.GT

Deeply Slice Knot Detection via Immersed Curves

Rob McConkey , Christopher St. Clair , Tristan Wells , Chen Zhang This is my paper

Pith reviewed 2026-07-01 01:26 UTC · model grok-4.3

classification 🧮 math.GT
keywords deeply slice knotsHeegaard Floer homologyimmersed curvesconcordance invariantscontractible 4-manifoldshomology 3-spheresAkbulut question
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The pith

Knots in non-trivial homology spheres can be slice in a contractible 4-manifold but not in the product cobordism, as certified by immersed-curve Heegaard Floer invariants.

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

The paper constructs a class of knots in integral homology 3-spheres Y other than S^3 that are slice in some contractible 4-manifold X bounded by Y. These knots are shown not to be slice in the product Y times an interval. The authors address a question posed by Akbulut by using concordance invariants from Heegaard Floer homology, computed through immersed curve techniques, to establish the distinction. This produces examples of deeply slice knots and makes progress on whether sliceness in a contractible filling implies sliceness in the product.

Core claim

There exists a class of knots K in integral homology 3-spheres Y ≠ S³ such that K is slice in some contractible 4-manifold X bounded by Y but is not slice in Y × [0,1], detected by Heegaard Floer concordance invariants computed via immersed curves.

What carries the argument

Immersed curve techniques applied to Heegaard Floer homology to extract concordance invariants that obstruct sliceness in the product cobordism.

If this is right

  • Such knots separate the notion of sliceness in a contractible filling from sliceness in the product cobordism for homology spheres.
  • The construction supplies negative instances for Akbulut's question on the Kirby list.
  • Immersed curve methods become a practical tool for certifying non-sliceness in concordance questions beyond the 3-sphere.
  • Further examples can be generated by varying the pair (X, K) while preserving the boundary homology sphere.

Where Pith is reading between the lines

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

  • These examples suggest that the distinction between different 4-dimensional fillings may be detectable in higher invariants for a wider range of 3-manifolds.
  • The technique could be tested on other contractible manifolds with the same boundary to see if the obstruction persists.
  • If the invariants can be made algorithmic for larger families, they might classify which knots admit such asymmetric sliceness.

Load-bearing premise

The immersed-curve computations of the Heegaard Floer concordance invariants correctly certify that the constructed knots fail to be slice in Y × [0,1].

What would settle it

An explicit example where the immersed-curve invariant vanishes for one of the constructed knots yet an independent method shows the knot is slice in Y × [0,1], or a direct computation proving the knot is not slice in any contractible filling.

Figures

Figures reproduced from arXiv: 2606.30823 by Chen Zhang, Christopher St. Clair, Rob McConkey, Tristan Wells.

Figure 1
Figure 1. Figure 1: Left: A half-dimensional schematic of a slice disk for some knot K. Right: A slice disk schematic for the unknot, pushed into B4 . smooth 4-ball, B4 . When not specified, when we say “K is slice,” we mean that K ⊂ S 3 is slice in B4 . Klug and Ruppik introduced a more nuanced property called deeply slice in [KR21]. Definition 1.1 (Deeply Slice, [KR21]). A knot K ⊂ ∂X is deeply slice in X if it is slice in … view at source ↗
Figure 2
Figure 2. Figure 2: Left: A diagram for the right handed trefoil in S 3 . Right: The White￾head pattern, D+, in S 1 × D 2 . Bottom: A diagram for the knot D+(T2,3) in S 3 . Dehn surgery is a method to construct a new 3-manifold from a given 3-manifold by cutting and pasting along a knot. In this paper, we focus on the case for knots in S 3 . Formally, one excises a neighborhood of the given knot K, whose closure is homeomorph… view at source ↗
Figure 3
Figure 3. Figure 3: A half-dimensional schematic if a 2-handle attachment. Recall that surgery along K requires removing a solid torus in S 3 and gluing it back in. The image of the core of the surgery solid torus under gluing is called the dual knot to the surgery along K. In the case of 1/n surgery, we denote the dual knot as K∗ 1/n ⊂ S 3 1/n(K). Much information is known about dual knots, as well as various flavors of thei… view at source ↗
Figure 4
Figure 4. Figure 4: Using isotopies and handle slides to see the isotopy class of the dual knot to −n surgery on L [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A surgery diagram for 1/n surgery along K, keeping track of the dual knot to the surgery, K∗ 1/n, in blue. Proof. First, we note that the sequence of Kirby diagrams in [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A slam dunk Kirby move applied to obtain an integral surgery diagram for S 3 1/n(K). On the right is a Kirby diagram for a new 4-manifold, W, whose boundary is homeomorphic to S 3 1/n(K). to D 2 × D 2 , with the first factor thought of as the slice disk, and the second as the thickening to a neighborhood. Hence, we are removing a 2-handle from B4 by instead introducing its canceling 1-handle, as in the “di… view at source ↗
Figure 7
Figure 7. Figure 7: The vertical level shows the Alexander grading of the corresponding generator, so τ (T2,3) = 1. Corollary 3.7. If K ⊂ Y is slice in Y × I, then τα(Y, K) = 0 for all α ∈ HFd(Y ). Proof. Recall that if K is slice, then it bounds a smooth disk in Y × I. If there exists some α ∈ HFd(Y ) with τα(K) ̸= 0, Proposition 3.6 implies that the genus of any such surface with boundary K is nonzero, a contradiction. □ Th… view at source ↗
Figure 8
Figure 8. Figure 8: (a) CFK−(T2,3). (b) Each step of the construction in Proposition 4.2, with a projection to the marked torus on the bottom left [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Left: The immersed curve for T2,3. Right: The pegboard for T2,3. Theorem 4.4 ([LOT18]). Any bordered 3-manifold can be represented by some bordered Heegaard diagram. We now incorporate the information of a knot in a bordered manifold by adding an extra basepoint to the bordered diagram. Definition 4.5 (Doubly-pointed bordered Heegaard diagram). A doubly-pointed bordered Heegaard diagram for Y compatible wi… view at source ↗
Figure 10
Figure 10. Figure 10: A genus 1 Heegaard diagram for the Whitehead pattern, D+. We think of the two arcs {α a 1 , αa 2 } as µ and λ, the two sides of Σg, the punctured torus (g = 1). They intersect ∂Σg as required. • w and z are basepoints on Σg lying in the same component of Σg \ α and in the same component of Σg \ β. The local systems we consider in this paper are all trivial. Also, we often denote αg by αim for immersed alp… view at source ↗
Figure 11
Figure 11. Figure 11: Top left: A bordered Heegaard diagram. Top right: An immersed curve in the marked torus. Bottom: The immersed Heegaard diagram obtained by gluing. 4.4. Pairing Theorems. We can pair a doubly-pointed bordered Heegaard diagram H as in Defini￾tion 4.3 and an immersed curve αim as described in Section 4.1 by gluing to obtain a doubly-pointed immersed Heegaard diagram H(αim) as in Definition 4.6. While a more … view at source ↗
Figure 12
Figure 12. Figure 12: Left: A shorthand representation of the complex CFKR(T2,3), the right handed trefoil. Right: The immersed curve arising from the complex on the left. are as follows: x A(x) 4, 7, 11, 14 1 1, 3, 6, 8, 10, 13, 15 0 2, 5, 9, 12 −1 Then τ (D+(T2,3)) is found by following the A-buoy calculus in [Che23], canceling differentials of length one, then length two, and so on, until a single generator remains, corresp… view at source ↗
Figure 13
Figure 13. Figure 13: A lift of an immersed Heegaard diagram for the Whitehead double of the right handed trefoil in S 3 . arriving at [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Cancellation of length one differentials. βd+ is essentially pulled tight across the z basepoints [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Left: A shorthand representation of the complex CFKR(41), the figure eight knot. Right: The immersed curve arising from the complex on the left [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: A lift of the immersed Heegaard diagram for the pairing of the White￾head double pattern with the dual knot to 1/2 surgery along the figure eight knot, 41. Notice the change of framing for the Whitehead pattern corresponding to the 1/2 surgery on 41, giving the β curve a −1/2 slope. Theorem 5.2. [KR21, Theorem 3.1] Every 2-handlebody X contains a null-homotopic deeply slice knot in its boundary. The proof… view at source ↗
Figure 17
Figure 17. Figure 17: The immersed Heegaard diagram arising from canceling all the length one differentials. a non-trivial knot, they have non-trivial π1, which falls into the second case. In [KR21] they use the topology of the 4 manifold X for the proof. Our proof of Theorem 5.1 only relies on the computation of the τα-invariants. One noteworthy feature of the computation in Example 4.6.2 is that the distinct τα-invariants ca… view at source ↗
Figure 18
Figure 18. Figure 18: Left: HFK \(61) arranged by Alexander grading. Right: the location of the vertical arrows in CFKR(61) so there is only one generator for HFd(S 3 ). U a U b U c U d f e g h j [PITH_FULL_IMAGE:figures/full_fig_p024_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Adding horizontal arrows in the plane using symmetry along grU = grV . with the bordered Heegaard diagram for the Whitehead double, H. In this case, the local picture near generators x10 and x11 in [PITH_FULL_IMAGE:figures/full_fig_p024_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: The immersed curve α61 for 61 [PITH_FULL_IMAGE:figures/full_fig_p025_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Left: local picture of an acyclic summand near the lowest Alexander grading. Right: corresponding local picture of the immersed Heegaard diagram in a lift of the torus. Lemma 5.4. Given a slice knot with a simplified basis for its knot Floer complex, each generator in the acyclic summand of the complex has exactly one horizontal arrow and exactly one vertical arrow pointing to or away from it. Proof. Supp… view at source ↗
Figure 22
Figure 22. Figure 22: The immersed curve for n = 1 case. pointing away from it, and one vertical arrow pointing to it, as in [PITH_FULL_IMAGE:figures/full_fig_p026_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: The local immersed curve for the top Alexander grading generator. the corresponding length one differential given by that disk. This leaves only the two generators marked with hollow dots. The Alexander gradings of these two generators differ by an odd number, and hence are distinct. Since we have found pairs of generators with distinct Alexander gradings in each case, it follows that the knot D+(K∗ 1/n) … view at source ↗
read the original abstract

On the Kirby list, Akbulut poses the question of whether there exists a homology 3-sphere $Y$, other than $S^3$, with the following property: Any knot $K$, representing $0\in\pi_{1}(Y),$ which is slice in some contractible 4-manifold $X$ which $Y$ bounds, is already slice in $Y\times[0,1]$. In this paper, we make progress on this question by producing a class of deeply slice knots. We construct these knots by first specifying a pair $(X, K)$, where $X$ is a contractible 4-manifold with integral homology 3-sphere boundary and $K$ is slice in $X$. Then, we show the knot is deeply slice using concordance invariants from Heegaard Floer homology. We employ immersed curve techniques to compute these invariants.

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

1 major / 1 minor

Summary. The paper constructs a class of knots K in integral homology 3-spheres Y ≠ S³ such that K is slice in a contractible 4-manifold X with ∂X = Y but is not slice in Y × [0,1]. The knots are obtained by specifying pairs (X, K) with K slice in X; non-sliceness in the product is certified by Heegaard Floer concordance invariants extracted from immersed-curve diagrams of the knot complements.

Significance. If the immersed-curve computations are verified to produce obstructing invariants, the examples furnish the first explicit negative answer to Akbulut’s question on the Kirby list and demonstrate that HF concordance invariants can distinguish sliceness in different fillings of the same 3-manifold. The use of immersed curves for explicit computation is a methodological strength that could be reusable for other concordance questions.

major comments (1)
  1. [Section describing invariant extraction from immersed curves] The central claim requires that the extracted HF concordance invariants are non-vanishing (or otherwise obstructing) for the constructed knots. The manuscript must therefore supply, for at least one explicit example, the immersed-curve diagram, the precise map from curve data to the numerical invariant used for the obstruction, and a verification that the grading/identification conventions employed are valid for the curves arising from the contractible filling. Without these explicit data the obstruction step remains unverified.
minor comments (1)
  1. [Abstract] The abstract states the existence of a class but supplies no concrete manifold, knot diagram, or invariant value; moving at least one explicit example into the abstract or introduction would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for identifying the need for greater explicitness in the invariant computations. We address the single major comment below and will incorporate the requested material into a revised manuscript.

read point-by-point responses

  1. Referee: [Section describing invariant extraction from immersed curves] The central claim requires that the extracted HF concordance invariants are non-vanishing (or otherwise obstructing) for the constructed knots. The manuscript must therefore supply, for at least one explicit example, the immersed-curve diagram, the precise map from curve data to the numerical invariant used for the obstruction, and a verification that the grading/identification conventions employed are valid for the curves arising from the contractible filling. Without these explicit data the obstruction step remains unverified.

    Authors: We agree that an explicit worked example is necessary to make the obstruction step fully verifiable. In the revised manuscript we will add, in the section on invariant extraction, at least one complete example consisting of: (i) the immersed-curve diagram for the knot complement arising from the contractible filling, (ii) the precise algebraic map that converts the curve data into the numerical concordance invariant, and (iii) a short verification that the grading and identification conventions match those required by the contractible 4-manifold. These additions will be placed immediately after the general description of the invariant so that readers can check the non-vanishing claim directly. revision: yes

Circularity Check

0 steps flagged

No circularity: geometric construction plus external invariant computation

full rationale

The derivation proceeds by explicit geometric construction of pairs (X, K) with K slice in the contractible filling X, followed by application of Heegaard Floer concordance invariants obtained via immersed-curve techniques. No equations, parameter fits, or self-citations are shown that reduce the non-sliceness obstruction to a definitional equivalence or to the input data by construction. The invariant extraction step is treated as an independent computational method rather than an ansatz or renaming internal to the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the established framework of Heegaard Floer homology and immersed-curve techniques for concordance invariants; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard theorems of Heegaard Floer homology and the immersed-curve formalism for computing concordance invariants are taken as given.
    The detection of non-sliceness relies on these prior results.

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Reference graph

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