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arxiv: 2606.02159 · v1 · pith:LPFCGR5Bnew · submitted 2026-06-01 · 🧮 math.GT

Minimal genus trisection diagrams of the elliptic surfaces E(n) via handle diagrams

Tsukasa Isoshima This is my paper

Pith reviewed 2026-06-28 11:43 UTC · model grok-4.3

classification 🧮 math.GT
keywords elliptic surfacestrisectionshandle diagramsLefschetz fibrations4-manifoldsminimal genus
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The pith

Elliptic surfaces E(n) admit explicit (12n-2,0)-trisection diagrams built directly from Lefschetz fibration handle diagrams.

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

Previous results showed that E(n) admits a minimal-genus (12n-2,0)-trisection by realizing it as a branched cover of S^2 x S^2. This paper supplies an explicit diagram by converting the handle decomposition that arises from the standard Lefschetz fibration of E(n). The conversion preserves the trisection parameters and the genus bound. A reader cares because concrete diagrams make it possible to compute trisection invariants, compare different 4-manifold presentations, and study how fibrations interact with trisections.

Core claim

We clarify a way to construct an explicit (12n-2,0)-trisection diagram of E(n) from its handle diagram arising from its Lefschetz fibration.

What carries the argument

Direct mapping from the 1- and 2-handles of the Lefschetz fibration handle diagram to the three handlebodies that define the (12n-2,0)-trisection.

If this is right

  • The resulting diagrams are minimal genus by the branched-cover argument.
  • Each E(n) now has a concrete set of curves that can be drawn and manipulated.
  • Trisection invariants of E(n) become computable from the Lefschetz data.
  • The construction works uniformly for every positive integer n.

Where Pith is reading between the lines

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

  • The same handle-to-trisection conversion may apply to other Lefschetz fibered 4-manifolds whose minimal trisection genus is already known.
  • One could compare the resulting diagrams with those obtained from other constructions such as Kirby diagrams or Heegaard splittings.
  • Explicit pictures might reveal relations between the monodromy of the fibration and the attaching curves of the trisection.

Load-bearing premise

The handle diagram coming from the Lefschetz fibration can be turned into a trisection diagram while keeping the genus at exactly 12n-2.

What would settle it

For E(1) or E(2), apply the conversion and obtain a diagram whose central surface has genus larger than 12n-2 or fails to satisfy the trisection gluing conditions.

Figures

Figures reproduced from arXiv: 2606.02159 by Tsukasa Isoshima.

Figure 1
Figure 1. Figure 1: A (1, 0)-trisection diagram of CP 2 . Remark 2.5. Given a genus g trisection whose spine is Hα ∪ Hβ ∪ Hγ, let α (resp. β, γ) be the boundary of meridian disk systems of Hα (resp. Hβ, Hγ). Then, (Σg; α, β, γ) is the trisection diagram with respect to the trisection. Conversely, given a trisection diagram (Σg; α, β, γ), by attaching 2-handles to Σg × D2 along α × {e 2πi 3 }, β × {e 4πi 3 } and γ × {e 2πi} wi… view at source ↗
Figure 2
Figure 2. Figure 2: Genus-1 trisection diagrams of S 4 . Theorem 2.8 ([GK16, Theorem 4, Theorem 11]). Every closed 4-manifold admits a trisection. Any two trisections of the same closed 4-manifold are isotopic after performing some number of stabilizations. Corollary 2.9 ([GK16, Corollary 12]). Two closed 4-manifolds are diffeomorphic if and only if two corresponding trisection diagrams are related by surface diffeo￾morphisms… view at source ↗
Figure 3
Figure 3. Figure 3: At Step 3 in the algorithm, each crossing is converted as described in this figure [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A handle diagram of E(n) obtained from its Lefschetz fibration (see [GS99, [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: After Step1 in the algorithm in Theorem 2.10 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: After Steps 2 and 3 in the algorithm in Theorem 2.10. The black −1 and −n omit the Dehn twist for γ￾curves as in [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Notations in Figures 6 and 26 for non-negative integers n. The bottom figure describes that we perform the Dehn twist t −n along the black curve for all γ-curves intersecting with the black curve (labeled the rectangle) [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: After Steps 4 and 5 in the algorithm in Theorem 2.10. Proof. In [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 12
Figure 12. Figure 12 [PITH_FULL_IMAGE:figures/full_fig_p007_12.png] view at source ↗
Figure 9
Figure 9. Figure 9: After Step 6 in the algorithm in Theorem 2.10. This is a (36n 2+6n+6; 2, 2, 36n 2−6n+4)-trisection diagram of E(n) [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: A (12n + 4, 2)-trisection diagram of E(n) obtained from [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: In this paper, handle slides are indicated by arrows as in this figure. If it is necessary to specify the or￾der of handle slides, we represent them by labeling arrows as in this figure. Namely, the most left fig￾ure means that firstly we perform the handle slide labeled 1 (the central figure), and then we perform the handle slide labeled 2 (the most right figure). We consider the following lemma before p… view at source ↗
Figure 13
Figure 13. Figure 13: A (12n + 4, 2)-trisection diagram of E(n) obtained from [PITH_FULL_IMAGE:figures/full_fig_p010_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: A (12n + 3; 2, 2, 1)-trisection diagram of E(n) ob￾tained from [PITH_FULL_IMAGE:figures/full_fig_p011_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: A (12n + 3; 2, 2, 1)-trisection diagram of E(n) ob￾tained from [PITH_FULL_IMAGE:figures/full_fig_p012_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: A (12n + 2; 2, 2, 0)-trisection diagram of E(n) ob￾tained from [PITH_FULL_IMAGE:figures/full_fig_p013_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: A (12n + 2; 2, 2, 0)-trisection diagram of E(n) ob￾tained from [PITH_FULL_IMAGE:figures/full_fig_p014_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: A (12n + 2; 2, 2, 0)-trisection diagram of E(n) ob￾tained from [PITH_FULL_IMAGE:figures/full_fig_p014_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: A (12n + 1; 1, 2, 0)-trisection diagram of E(n) ob￾tained from [PITH_FULL_IMAGE:figures/full_fig_p015_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: A (12n + 1; 1, 2, 0)-trisection diagram of E(n) ob￾tained from [PITH_FULL_IMAGE:figures/full_fig_p015_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: A (12n; 0, 2, 0)-trisection diagram of E(n) obtained from [PITH_FULL_IMAGE:figures/full_fig_p016_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: A destabilization for a Heegaard diagram. If β ′ 1, . . . , β′ n are parallel to γ ′ 1, . . . , γ′ n, then β1, . . . , βn can be par￾allel to γ1, . . . , γn by handle slides over βi [PITH_FULL_IMAGE:figures/full_fig_p016_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Handle slides for β-curves in Lemma 3.6 [PITH_FULL_IMAGE:figures/full_fig_p016_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: A (12n; 0, 2, 0)-trisection diagram of E(n) obtained from [PITH_FULL_IMAGE:figures/full_fig_p017_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: A (12n; 0, 2, 0)-trisection diagram of E(n) obtained from [PITH_FULL_IMAGE:figures/full_fig_p017_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: A (12n; 0, 2, 0)-trisection diagram of E(n) obtained from [PITH_FULL_IMAGE:figures/full_fig_p018_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: A (12n; 0, 2, 0)-trisection diagram of E(n) obtained from [PITH_FULL_IMAGE:figures/full_fig_p018_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: A (12n; 0, 2, 0)-trisection diagram of E(n) obtained from [PITH_FULL_IMAGE:figures/full_fig_p019_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: A (12n; 0, 2, 0)-trisection diagram of E(n) obtained from [PITH_FULL_IMAGE:figures/full_fig_p019_29.png] view at source ↗
read the original abstract

Lambert-Cole and Meier showed that the elliptic surface $E(n)$ admits a $(12n-2,0)$-trisection, considering the property that $E(n)$ is a certain double branched cover of $S^2 \times S^2$, which is a minimal genus trisection. In this paper, we clarify a way to construct an explicit $(12n-2,0)$-trisection diagram of $E(n)$ from its handle diagram arising from its Lefschetz fibration.

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 claims to clarify an explicit construction that converts the handle diagram of the Lefschetz fibration of the elliptic surface E(n) into a (12n-2,0)-trisection diagram. Minimality of the genus is inherited from the branched-cover argument of Lambert-Cole and Meier rather than reproved here.

Significance. If the conversion steps are correct and fully explicit, the work supplies concrete trisection diagrams for a fundamental family of 4-manifolds. Explicit, handle-diagram-based constructions are a strength that can support further computations in trisection theory.

minor comments (2)
  1. [Abstract] Abstract: the description of the construction is high-level; a sentence outlining the main conversion steps (e.g., how curves or handles are mapped) would improve accessibility without lengthening the abstract.
  2. The manuscript would benefit from a short example (e.g., n=1 or n=2) showing the handle diagram before and after conversion to illustrate the method concretely.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review and recommendation of minor revision. The referee's summary correctly describes the paper's contribution as providing an explicit conversion from the Lefschetz fibration handle diagram to a (12n-2,0)-trisection diagram, with minimality inherited from Lambert-Cole and Meier.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper supplies an explicit construction that converts a handle diagram (arising from the Lefschetz fibration of E(n)) into a (12n-2,0)-trisection diagram. Minimality of the genus is inherited from an external citation to Lambert-Cole and Meier (different authors) whose branched-cover argument is independent of the present construction steps. No self-citation is load-bearing, no parameter is fitted and then renamed as a prediction, and no equation or diagram count reduces to its own input by definition. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper builds on the existence result from prior work without introducing new free parameters or entities based on the abstract.

axioms (1)
  • domain assumption E(n) admits a (12n-2,0)-trisection as a double branched cover of S^2 × S^2
    This is the property used by Lambert-Cole and Meier that the paper builds upon.

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Reference graph

Works this paper leans on

147 extracted references · 71 canonical work pages · 3 internal anchors

  1. [1]

    and Stipsicz, Andr\'as I

    Gompf, Robert E. and Stipsicz, Andr\'as I. , TITLE =. 1999 , PAGES =. doi:10.1090/gsm/020 , URL =

    work page doi:10.1090/gsm/020 1999

  2. [2]

    Topology , FJOURNAL =

    Waldhausen, Friedhelm , TITLE =. Topology , FJOURNAL =. 1968 , PAGES =. doi:10.1016/0040-9383(68)90027-X , URL =

    work page doi:10.1016/0040-9383(68)90027-x 1968

  3. [3]

    2016 , school=

    Relative trisections of smooth 4-manifolds with boundary , author=. 2016 , school=

    2016

  4. [4]

    and Gay, David T

    Castro, Nickolas A. and Gay, David T. and Pinz\'. Diagrams for relative trisections , JOURNAL =. 2018 , NUMBER =

    2018

  5. [5]

    and Ozbagci, Burak , TITLE =

    Castro, Nickolas A. and Ozbagci, Burak , TITLE =. Math. Res. Lett. , FJOURNAL =. 2019 , NUMBER =

    2019

  6. [6]

    Gay, David and Kirby, Robion , TITLE =. Geom. Topol. , FJOURNAL =. 2016 , NUMBER =

    2016

  7. [7]

    Mathematical Proceedings of the Cambridge Philosophical Society , volume=

    Doubly pointed trisection diagrams and surgery on 2-knots , author=. Mathematical Proceedings of the Cambridge Philosophical Society , volume=. 2022 , organization=

    2022

  8. [8]

    1999 , publisher=

    4-manifolds and Kirby calculus , author=. 1999 , publisher=

    1999

  9. [9]

    Kim, Seungwon and Miller, Maggie , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2020 , NUMBER =

    2020

  10. [10]

    Michigan Math

    Katanaga, Atsuko and Saeki, Osamu and Teragaito, Masakazu and Yamada, Yuichi , TITLE =. Michigan Math. J. , FJOURNAL =. 1999 , NUMBER =

    1999

  11. [11]

    A note on

    Laudenbach, Fran. A note on. Bull. Soc. Math. France , FJOURNAL =. 1972 , PAGES =

    1972

  12. [12]

    Pacific Journal of Mathematics , volume=

    Proof of a conjecture of Whitney , author=. Pacific Journal of Mathematics , volume=. 1969 , publisher=

    1969

  13. [13]

    Meier, Jeffrey , TITLE =. Math. Res. Lett. , FJOURNAL =. 2018 , NUMBER =

    2018

  14. [14]

    Meier, Jeffrey and Schirmer, Trent and Zupan, Alexander , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2016 , NUMBER =

    2016

  15. [15]

    Meier, Jeffrey and Zupan, Alexander , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2017 , NUMBER =

    2017

  16. [16]

    Meier, Jeffrey and Zupan, Alexander , TITLE =. Proc. Natl. Acad. Sci. USA , FJOURNAL =. 2018 , NUMBER =

    2018

  17. [17]

    2019 , note=

    Trisection diagrams and twists of 4-manifolds , author=. 2019 , note=

    2019

  18. [18]

    Price, T. M. , TITLE =. J. Austral. Math. Soc. Ser. A , FJOURNAL =. 1977 , NUMBER =

    1977

  19. [19]

    Workshop on

    Schleimer, Saul , TITLE =. Workshop on. 2007 , MRCLASS =. doi:10.2140/gtm.2007.12.299 , URL =

    work page doi:10.2140/gtm.2007.12.299 2007

  20. [20]

    Lectures in Topology , year=

    On the topology of differentiable manifolds , author=. Lectures in Topology , year=

  21. [21]

    Diagrams of *-Trisections , publisher =

    Aranda, Jos\'. Diagrams of *-Trisections , publisher =. 2019 , copyright =. doi:10.48550/ARXIV.1911.06467 , url =

    work page doi:10.48550/arxiv.1911.06467 2019

  22. [22]

    2024 , eprint=

    Bridge Multisections of Knotted Surfaces in S^4 , author=. 2024 , eprint=

    2024

  23. [23]

    2025 , eprint=

    Trisected Rainbows and Braids , author=. 2025 , eprint=

    2025

  24. [24]

    and Kirby, Robion , TITLE =

    Abrams, Aaron and Gay, David T. and Kirby, Robion , TITLE =. Geom. Topol. , FJOURNAL =. 2018 , NUMBER =

    2018

  25. [25]

    Tri-plane diagrams for simple surfaces in S^4 , publisher =

    Allred, Wolfgang and Arag\'. Tri-plane diagrams for simple surfaces in S^4 , publisher =. 2023 , copyright =. doi:10.48550/ARXIV.2302.08431 , url =

    work page doi:10.48550/arxiv.2302.08431 2023

  26. [26]

    Minimal genus four-manifolds , JOURNAL =

    Aranda Cuevas, Jos\'. Minimal genus four-manifolds , JOURNAL =. 2020 , NUMBER =. doi:10.1090/proc/14689 , URL =

    work page doi:10.1090/proc/14689 2020

  27. [27]

    Thin position through the lens of trisections of 4-manifolds , JOURNAL =

    Aranda, Rom\'. Thin position through the lens of trisections of 4-manifolds , JOURNAL =. 2021 , NUMBER =. doi:10.2140/pjm.2021.313.293 , URL =

    work page doi:10.2140/pjm.2021.313.293 2021

  28. [28]

    2021 , PAGES =

    Aranda Cuevas, Jose Roman , TITLE =. 2021 , PAGES =

    2021

  29. [29]

    Bounding the Kirby-Thompson invariant of spun knots

    Aranda, Rom\'. Bounding the Kirby-Thompson invariant of spun knots , publisher =. 2021 , copyright =. doi:10.48550/ARXIV.2112.02420 , url =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2112.02420 2021

  30. [30]

    2023 , eprint=

    Pants distances of knotted surfaces in 4-manifolds , author=. 2023 , eprint=

    2023

  31. [31]

    Manifolds with weakly reducible genus-three trisections are standard , Year =

    Aranda, Rom. Manifolds with weakly reducible genus-three trisections are standard , Year =

  32. [32]

    2023 , eprint=

    Multisections of higher-dimensional manifolds , author=. 2023 , eprint=

    2023

  33. [33]

    Right-left equivalent maps of simplified (2, 0) -trisections with different configurations of vanishing cycles , publisher =

    Asano, Nobutaka , keywords =. Right-left equivalent maps of simplified (2, 0) -trisections with different configurations of vanishing cycles , publisher =. 2021 , copyright =. doi:10.48550/ARXIV.2109.13533 , url =

    work page doi:10.48550/arxiv.2109.13533 2021

  34. [34]

    Vertical 3-manifolds in simplified (2, 0)-trisections of 4-manifolds , publisher =

    Asano, Nobutaka , keywords =. Vertical 3-manifolds in simplified (2, 0)-trisections of 4-manifolds , publisher =. 2020 , copyright =. doi:10.48550/ARXIV.2010.08239 , url =

    work page doi:10.48550/arxiv.2010.08239 2020

  35. [35]

    Some lower bounds for the Kirby-Thompson invariant , publisher =

    Asano, Nobutaka and Naoe, Hironobu and Ogawa, Masaki , keywords =. Some lower bounds for the Kirby-Thompson invariant , publisher =. 2023 , copyright =. doi:10.48550/ARXIV.2302.07447 , url =

    work page doi:10.48550/arxiv.2302.07447 2023

  36. [36]

    Gay and Peter Lambert-Cole , year=

    Sarah Blackwell and David T. Gay and Peter Lambert-Cole , year=. Constructing. 2306.16404 , archivePrefix=

    arXiv

  37. [37]

    A note on three-fold branched covers of S^4 , publisher =

    Blair, Ryan and Cahn, Patricia and Kjuchukova, Alexandra and Meier, Jeffrey , keywords =. A note on three-fold branched covers of S^4 , publisher =. 2019 , copyright =. doi:10.48550/ARXIV.1909.11788 , url =

    work page doi:10.48550/arxiv.1909.11788 2019

  38. [38]

    and Kawana, N

    Bannai, S. and Kawana, N. and Masuya, R. and Tokunaga, H. , TITLE =. Geom. Dedicata , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s10711-021-00672-5 , URL =

    work page doi:10.1007/s10711-021-00672-5 2022

  39. [39]

    Bell, Mark and Hass, Joel and Rubinstein, Joachim Hyam and Tillmann, Stephan , TITLE =. Proc. Natl. Acad. Sci. USA , FJOURNAL =. 2018 , NUMBER =. doi:10.1073/pnas.1717173115 , URL =

    work page doi:10.1073/pnas.1717173115 2018

  40. [40]

    Baykur, R. \. Simplified broken. Proc. Natl. Acad. Sci. USA , FJOURNAL =. 2018 , NUMBER =. doi:10.1073/pnas.1717175115 , URL =

    work page doi:10.1073/pnas.1717175115 2018

  41. [41]

    Simplifying indefinite fibrations on 4-manifolds

    Baykur, R. Inanc and Saeki, Osamu , keywords =. Simplifying indefinite fibrations on 4-manifolds , publisher =. 2017 , copyright =. doi:10.48550/ARXIV.1705.11169 , url =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1705.11169 2017

  42. [42]

    and Tomova, Maggy , TITLE =

    Blair, Ryan and Campisi, Marion and Taylor, Scott A. and Tomova, Maggy , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2022 , NUMBER =. doi:10.1112/jlms.12513 , URL =

    work page doi:10.1112/jlms.12513 2022

  43. [43]

    2025 , HowPublished =

    Bridges, Nicolas and Cui, Shawn , Title =. 2025 , HowPublished =

    2025

  44. [44]

    Singular branched covers of four-manifolds

    Cahn, Patricia and Kjuchukova, Alexandra , keywords =. Singular branched covers of four-manifolds , publisher =. 2017 , copyright =. doi:10.48550/ARXIV.1710.11562 , url =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1710.11562 2017

  45. [45]

    2023 , eprint=

    Algorithms for Computing Invariants of Trisected Branched Covers , author=. 2023 , eprint=

    2023

  46. [46]

    2025 , eprint=

    Kirby diagrams, trisections and gems of PL 4-manifolds: relationships, results and open problems , author=. 2025 , eprint=

    2025

  47. [47]

    2025 , howpublished =

    Casali, Maria Rita and Cristofori, Paola , title =. 2025 , howpublished =

    2025

  48. [48]

    2025 , eprint=

    Categorical 4-manifold invariants from trisection diagrams , author=. 2025 , eprint=

    2025

  49. [49]

    and Gay, David T

    Castro, Nickolas A. and Gay, David T. and Pinz\'. Trisections of 4-manifolds with boundary , JOURNAL =. 2018 , NUMBER =

    2018

  50. [50]

    The contact cut graph and a Weinstein

    Nickolas Castro and Gabriel Islambouli and Jie Min and Sümeyra Sakallı and Laura Starkston and Angela Wu , year=. The contact cut graph and a Weinstein. 2408.05340 , archivePrefix=

    arXiv

  51. [51]

    2024 , eprint=

    Relative Group Trisections , author=. 2024 , eprint=

    2024

  52. [52]

    and Ozbagci, Burak , TITLE =

    Castro, Nickolas A. and Ozbagci, Burak , TITLE =. Math. Res. Lett. , FJOURNAL =. 2019 , NUMBER =. doi:10.4310/MRL.2019.v26.n2.a3 , URL =

    work page doi:10.4310/mrl.2019.v26.n2.a3 2019

  53. [53]

    2023 , eprint=

    Combed Trisection Diagrams and Non-Semisimple 4-Manifold Invariants , author=. 2023 , eprint=

    2023

  54. [54]

    Chu, Michelle and Tillmann, Stephan , TITLE =. Rev. Roumaine Math. Pures Appl. , FJOURNAL =. 2019 , NUMBER =

    2019

  55. [55]

    2023 , eprint=

    Multisections of surface bundles and bundles over the circle , author=. 2023 , eprint=

    2023

  56. [56]

    2024 , eprint=

    Multisections of (m+3) -dimensional m -spun 3 -manifolds , author=. 2024 , eprint=

    2024

  57. [57]

    2023 , eprint=

    Relative trisections of fiber bundles over the circle , author=. 2023 , eprint=

    2023

  58. [58]

    Feller, Peter and Klug, Michael and Schirmer, Trenton and Zemke, Drew , TITLE =. Proc. Natl. Acad. Sci. USA , FJOURNAL =. 2018 , NUMBER =

    2018

  59. [59]

    Florens, Vincent and Moussard, Delphine , TITLE =. Canad. J. Math. , FJOURNAL =. 2022 , NUMBER =. doi:10.4153/S0008414X20000863 , URL =

    work page doi:10.4153/s0008414x20000863 2022

  60. [60]

    2024 , eprint=

    Pseudo-trisections of four-manifolds with boundary , author=. 2024 , eprint=

    2024

  61. [61]

    , TITLE =

    Gay, David T. , TITLE =. Winter Braids Lect. Notes , FJOURNAL =. 2018 , NUMBER =

    2018

  62. [62]

    From near-symplectic constructions to trisections of 4-manifolds , publisher =

    Gay, David T , keywords =. From near-symplectic constructions to trisections of 4-manifolds , publisher =. 2022 , copyright =. doi:10.48550/ARXIV.2206.07686 , url =

    work page doi:10.48550/arxiv.2206.07686 2022

  63. [63]

    , TITLE =

    Gay, David T. , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2016 , NUMBER =

    2016

  64. [64]

    Gay, David and Meier, Jeffrey , TITLE =. Math. Proc. Cambridge Philos. Soc. , FJOURNAL =. 2022 , NUMBER =. doi:10.1017/S0305004121000165 , URL =

    work page doi:10.1017/s0305004121000165 2022

  65. [65]

    , TITLE =

    Ghanwat, Abhijeet and Pandit, Suhas and Selvakumar, A. , TITLE =. J. Knot Theory Ramifications , FJOURNAL =. 2024 , NUMBER =. doi:10.1142/S0218216524500111 , URL =

    work page doi:10.1142/s0218216524500111 2024

  66. [66]

    Rocky Mountain J

    Ghanwat, Abhijeet and Pandit, Suhas and A., Selvakumar , TITLE =. Rocky Mountain J. Math. , FJOURNAL =. 2026 , NUMBER =. doi:10.1216/rmj.2026.56.149 , URL =

    work page doi:10.1216/rmj.2026.56.149 2026

  67. [67]

    Moduli Spaces of Lagrangian Surfaces in

    Devashi Gulati and Peter Lambert-Cole , year=. Moduli Spaces of Lagrangian Surfaces in. 2406.12767 , archivePrefix=

    arXiv

  68. [68]

    Hayano, Kenta , TITLE =. Osaka J. Math. , FJOURNAL =. 2020 , NUMBER =

    2020

  69. [69]

    Band diagrams of immersed surfaces in 4-manifolds , publisher =

    Hughes, Mark and Kim, Seungwon and Miller, Maggie , keywords =. Band diagrams of immersed surfaces in 4-manifolds , publisher =. 2021 , copyright =. doi:10.48550/ARXIV.2108.12794 , url =

    work page doi:10.48550/arxiv.2108.12794 2021

  70. [70]

    Islambouli, Gabriel , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2018 , NUMBER =

    2018

  71. [71]

    Islambouli, Gabriel , TITLE =. Geom. Dedicata , FJOURNAL =. 2021 , PAGES =. doi:10.1007/s10711-021-00617-y , URL =

    work page doi:10.1007/s10711-021-00617-y 2021

  72. [72]

    2021 , eprint=

    Uniqueness of 4-manifolds described as sequences of 3-d handlebodies , author=. 2021 , eprint=

    2021

  73. [73]

    Multisections of 4-manifolds , publisher =

    Islambouli, Gabriel and Naylor, Patrick , keywords =. Multisections of 4-manifolds , publisher =. 2020 , copyright =. doi:10.48550/ARXIV.2010.03057 , url =

    work page doi:10.48550/arxiv.2010.03057 2020

  74. [74]

    Islambouli, Gabriel and Naylor, Patrick , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2024 , NUMBER =. doi:10.1090/tran/8996 , URL =

    work page doi:10.1090/tran/8996 2024

  75. [75]

    2023 , eprint=

    Multisections with divides and Weinstein 4-manifolds , author=. 2023 , eprint=

    2023

  76. [76]

    2022 , eprint=

    Toric multisections and curves in rational surfaces , author=. 2022 , eprint=

    2022

  77. [77]

    Representing smooth 4-manifolds as loops in the pants complex , publisher =

    Islambouli, Gabriel and Klug, Michael , keywords =. Representing smooth 4-manifolds as loops in the pants complex , publisher =. 2019 , copyright =. doi:10.48550/ARXIV.1912.02325 , url =

    work page doi:10.48550/arxiv.1912.02325 2019

  78. [78]

    Trisections obtained by trivially regluing surface-knots , publisher =

    Isoshima, Tsukasa , keywords =. Trisections obtained by trivially regluing surface-knots , publisher =. 2022 , copyright =. doi:10.48550/ARXIV.2205.04817 , url =

    work page doi:10.48550/arxiv.2205.04817 2022

  79. [79]

    Isoshima, Tsukasa , TITLE =. Geom. Dedicata , FJOURNAL =. 2024 , NUMBER =. doi:10.1007/s10711-024-00919-x , URL =

    work page doi:10.1007/s10711-024-00919-x 2024

  80. [80]

    Trisections induced by the

    Tsukasa Isoshima and Masaki Ogawa , year=. Trisections induced by the. 2305.12042 , archivePrefix=

    arXiv

Showing first 80 references.