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arxiv: 2604.07684 · v1 · submitted 2026-04-09 · 🧮 math.GT

Kirby diagrams for an infinite family of exotic mathbb{R}⁴'s

Siddharth Shrivastava This is my paper

Pith reviewed 2026-05-10 18:18 UTC · model grok-4.3

classification 🧮 math.GT
keywords exotic R^4Kirby diagramCasson handleribbon knotpretzel knotslice disk complement4-manifolds
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The pith

Kirby diagrams are supplied for an infinite family of exotic R^4's from ribbon knots and pretzel knots.

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

The paper extends the construction of exotic R^4's by Eli, Hom, and Lidman. It supplies explicit Kirby diagrams for the case of the ribbon knot T_{2,n} # T_{2,-n} when n is odd and at least 3. It proves that two such families of diagrams are equivalent and then generalizes the diagrams to a family using the pretzel knots P(n, -n, 2k). This matters because it makes these exotic structures more accessible for study through concrete diagrams rather than abstract existence.

Core claim

By providing Kirby diagrams for attaching the simplest positive Casson handle to the slice disc complements of the knots T_{2,n}#T_{2,-n} for odd n≥3 and showing equivalence of two families, the paper makes these exotic R^4's explicit. It further generalizes the diagrams to ribbon disc complements of the pretzel knots P(n,-n,2k).

What carries the argument

Kirby diagrams for the Casson handle attachment to slice disc complements of ribbon knots.

Load-bearing premise

That the diagrams accurately represent the attachment of the simplest positive Casson handle to the slice disc complements of these knots for general odd n and pretzel knots.

What would settle it

Verifying whether the generalized diagram for a specific pretzel knot produces a manifold that is diffeomorphic to standard R^4 or not homeomorphic to R^4 would test the claim.

Figures

Figures reproduced from arXiv: 2604.07684 by Siddharth Shrivastava.

Figure 1
Figure 1. Figure 1: A Kirby diagram for Rn. The diagram includes a total of n−1 dotted circles inside the large dotted circle. 0 n − 1 2 full windings 0 0 0 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A Kirby diagram for R′ n . The dashed lines represent (n − 1)/2 full windings around the inner strands of the two dotted circles. As a result, the two families of Kirby diagrams in Theorem 1.1 represent the same family of exotic R 4 ’s. Theorem 1.2. The manifolds Rn and R′ n are diffeomorphic for all n ≥ 3 and odd. Thus, {Rn}n=3,5,7,... and {R′ n}n=3,5,7,... are the same family of exotic R 4 ’s. Remark 1.3… view at source ↗
Figure 3
Figure 3. Figure 3: An equivalent Kirby diagram to [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A Kirby diagram for Rn,k. The box represents k full twists in the two parallel strands of the 2-handle. Note. We have been informed that the Kirby diagrams in [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: a is the torus knot T2,5. The knot T2,−5 is the mirror image of T2,5, and so the connected sum T2,5#T2,−5 is the knot shown in Figure 5b and Figure 5c. Since torus knots are invertible [BZ03, Proposition 3.27], we do not need to consider orientations when taking the connected sum. (a) (b) (c) [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Ribbon moves for a ribbon disc complement of the knot T2,5#T2,−5. All 2-handles in (b) are 0-framed. We will now obtain an exotic R 4 by attaching a Casson handle to the ribbon disc complement shown in Figure 6c. Casson handles are built using layers of self-plumbed 2-handles. In general, [PITH_FULL_IMAGE:figures/full_fig_p004_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A Kirby diagram for R5. To see how this diagram generalises for all n ≥ 3 and odd, we notice that the knot T2,n will differ from Figure 5a by having n half-twists, and so just like in Figure 6a, we will need to do n − 1 ribbon moves. As we saw in Figure 6b, this will produce n unlinked dotted circles, with n − 1 0-framed 2-handles as a result of the ribbon moves. Similar to Figure 6c, this will yield the r… view at source ↗
Figure 8
Figure 8. Figure 8: (a) The torus knot T2,5. (b) and (c) The connected sum T2,5#T2,−5. (a) 0 (b) 0 (c) 0 (d) [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Ribbon moves for a ribbon disc complement of the knot T2,5#T2,−5. 0 0 0 0 [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: A Kirby diagram for R′ 5 . of the 2-handle has 2n + 2 crossings with the left dotted circle, and 2n crossings with the right dotted circle. The case of n = 3 gives the same ribbon disc complement provided in [GS99, [PITH_FULL_IMAGE:figures/full_fig_p006_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Diagrams of two moves that result from handle slides and cancella￾tions. The vertical lines in Figure 11b represent any 2-handles that may be going through the dotted circles. As mentioned in Remark 1.3, we can represent Rn using the Kirby diagram in [PITH_FULL_IMAGE:figures/full_fig_p007_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The result of Kirby moves, going from the disc complement of Rn to the disc complement of R′ n . The diagram in (a) is the same as the disc complement in [PITH_FULL_IMAGE:figures/full_fig_p008_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Equivalent diagrams to Figure 12c. In (a), there are n−2 half-twists on both sides. In (b), there are n half-twists on both sides. In the general case for n ≥ 3 and odd, the knot P(n, −n, 2k) will differ from Figure 14a by having n positive half-twists on the left side and n negative half-twists on the right side. By doing the same single ribbon move, we will obtain a diagram similar to Figure 14d where t… view at source ↗
Figure 14
Figure 14. Figure 14: Ribbon moves for a ribbon disc complement of the knot P(5, −5, 2k). To prove the second part of the theorem, we first show that the knot Floer homology HFK \ of the knot P(n, −n, 2k) for n ≥ 3 and odd, and k ∈ Z, does not depend on k. We use a result [PITH_FULL_IMAGE:figures/full_fig_p009_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: a for the case n = 5. This is because by unwinding the band, as shown in Figure 15b, we get the same diagram as Figure 8c. In general, for n ≥ 3 and odd, we can obtain a similar diagram to Figure 15a for T2,n#T2,−n in which the band has (n − 1)/2 full windings around the two unknots. By comparing Figure 14a with Figure 15b, we can see that the knot P(n, −n, 2k) is obtained by adding k full twists to the b… view at source ↗
read the original abstract

Eli, Hom, and Lidman showed that the manifolds produced by attaching the simplest positive Casson handle $CH^+$ to a slice disc complement of the ribbon knot $T_{2,n}\#T_{2,-n}$ for $n\ge3$ and odd, and removing the boundary, form a countably infinite family of exotic $\mathbb{R}^4$'s. They provided a Kirby diagram for the case $n=3$. In this short note, we extend this for $n\ge3$ and odd, and provide Kirby diagrams for two such families of exotic $\mathbb{R}^4$'s, which are then shown to be equivalent. We then generalise these diagrams to a family of exotic $\mathbb{R}^4$'s built using ribbon disc complements of the pretzel knots $P(n,-n,2k)$.

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 / 2 minor

Summary. The manuscript extends the Eli-Hom-Lidman construction of exotic R^4's by attaching the simplest positive Casson handle CH^+ to the slice disc complement of the ribbon knot T_{2,n} # T_{2,-n} (n ≥ 3 odd). It supplies explicit Kirby diagrams for two such families, proves their equivalence via Kirby calculus moves, and generalizes the diagrams to a family built from ribbon disc complements of the pretzel knots P(n, -n, 2k).

Significance. If the diagrams accurately capture the Casson-handle attachments and the equivalence holds, the work supplies concrete, computable representations of an infinite family of exotic R^4's. This facilitates direct diagrammatic study and potential further calculations. The generalization to pretzel knots broadens the known examples while resting on the established exoticness result of Eli-Hom-Lidman; the absence of free parameters or ad-hoc axioms in the diagrammatic extensions is a strength.

minor comments (2)
  1. The description of the two families in the introduction would benefit from an explicit statement of how the second family differs diagrammatically from the first before the equivalence proof is given.
  2. In the generalization section, the notation for the pretzel knot parameters (n, -n, 2k) could be clarified with a brief reminder of the ribbon disc complement construction to aid readers unfamiliar with the specific family.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance in providing explicit Kirby diagrams for an infinite family of exotic R^4's, and recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's core contribution consists of explicit Kirby diagram constructions for families of exotic R^4's (extending the n=3 case from Eli-Hom-Lidman to odd n≥3, proving equivalence of two such families, and generalizing to pretzel knot complements P(n,-n,2k)). These steps rely on direct diagrammatic manipulation and the external Eli-Hom-Lidman theorem for the exoticness property itself. No load-bearing step reduces by definition, by fitting, or by self-citation chain to the paper's own inputs; the cited exoticness result is independent and externally established. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the external theorem of Eli, Hom, and Lidman for the exoticness property and on standard axioms of 4-manifold handle theory; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Eli, Hom, and Lidman's result that attaching the simplest positive Casson handle to the slice disc complement of T_{2,n}#T_{2,-n} (odd n>=3) yields an exotic R^4 after boundary removal
    The paper invokes this theorem to assert that the diagrams represent exotic R^4's without reproving the exoticness.
  • standard math Standard rules of Kirby calculus for handle attachments and diagram equivalences in 4-manifolds
    Used to show equivalence of the two families and to draw the generalized diagrams.

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Reference graph

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6 extracted references · 6 canonical work pages

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