Bridge trisections in ℂℙ2 and the Thom
conjecture

Lambert-Cole, Peter

doi:10.2140/gt.2020.24.1571

Cited by 15 publications

(11 citation statements)

References 17 publications

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“…Then the following "tropical" decomposition is in fact a (1, 0) trisection of X = CP 2 , and verifying this is also a good exercise: This trisection is used in [8] to give a combinatorial proof of the Thom conjecture (that algebraic curves in CP 2 minimize genus in their homology classes), and is also used in [9] to produce minimal genus trisections of a large class of algebraic surfaces.…”

Section: Iv-6mentioning

confidence: 98%

From Heegaard splittings to trisections; porting 3-dimensional ideas to dimension 4

T.¹

2020

Winter Braids Lecture Notes

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These notes summarize and expand on a mini-course given at CIRM in February 2018 as part of Winter Braids VIII. We somewhat obsessively develop the slogan "Trisections are to 4-manifolds as Heegaard splittings are to 3-manifolds", focusing on and clarifying the distinction between three ways of thinking of things: the basic definitions as decompositions of manifolds, the Morse theoretic perspective and descriptions in terms of diagrams. We also lay out these themes in two important relative settings: 4-manifolds with boundary and 4-manifolds with embedded 2dimensional submanifolds.

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Section: Iv-6mentioning

confidence: 98%

From Heegaard splittings to trisections; porting 3-dimensional ideas to dimension 4

T.¹

2020

Winter Braids Lecture Notes

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“…This trisection is compatible with the toric structure on

{CP}^{2}

, and Figure 1 shows the decomposition in the image of the moment map. The corners of

\partial Z_{λ}

can be smoothed by approximating

\partial Z_{λ}

f_{λ, N}^{- 1} false(1 false)

, where

f_{λ, N} false(z_{λ}, z_{λ + 1} false) : = \frac{1}{N} false(false| z_{λ} {false|}^{2} + false| z_{λ + 1} {false|}^{2} false) + false| z_{λ} {false|}^{2 N} + {| z_{λ + 1} |}^{2 N}

for

N > > 0

(as in [20]).…”

Section: Definitions Notions Of Equivalence and Open Questionsmentioning

confidence: 99%

“…The standard trisection of

{CP}^{2}

(see Example 2.3) can be constructed via toric geometry and therefore interacts nicely with the symplectic geometry. The contact type property of the boundaries of the three sectors in

{CP}^{2}

played an important role in the trisection‐theoretic proof of the Thom conjecture [20]. Further progress‐relating trisections and symplectic structures was made by Gay: Every closed symplectic manifold admits a Lefschetz pencil, and Gay described how to construct a trisection from a pencil [11]; see also related work on Lefschetz fibrations by Baykur‐Saeki [4] and Castro‐Ozbagci [7].…”

Section: Introductionmentioning

confidence: 99%

Symplectic 4‐manifolds admit Weinstein trisections

Lambert-Cole

Meier

Starkston

2021

Journal of Topology

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We prove that every symplectic 4‐manifold admits a trisection that is compatible with the symplectic structure in the sense that the symplectic form induces a Weinstein structure on each of the three sectors of the trisection. Along the way, we show that a (potentially singular) symplectic braided surface in CP2 can be symplectically isotoped into bridge position. This paper relies extensively on colour figures. Some references to colour may not be meaningful in the printed version, and we refer the reader to the online version which includes the colour figures.

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“…They also give rise to certain diagrammatic representations of knotted surfaces. In recent years, many authors have connected (bridge) trisections to major open problems in the theory of 2-knots and 4-manifolds [7,11,12].…”

Section: Introductionmentioning

confidence: 99%

Bounding the Kirby-Thompson invariant of spun knots

Román¹,

Pongtanapaisan²,

Taylor³

et al. 2021

Preprint

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A bridge trisection of a smooth surface in S 4 is a decomposition analogous to a bridge splitting of a link in S 3 . The Kirby-Thompson invariant of a bridge trisection measures its complexity in terms of distances between disc sets in the pants complex of the trisection surface. We give the first significant bounds for the Kirby-Thompson invariant of spun knots. In particular, we show that the Kirby-Thompson invariant of the spun trefoil is 15.

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Bridge trisections in ℂℙ2 and the Thom conjecture

Cited by 15 publications

References 17 publications

From Heegaard splittings to trisections; porting 3-dimensional ideas to dimension 4

From Heegaard splittings to trisections; porting 3-dimensional ideas to dimension 4

Symplectic 4‐manifolds admit Weinstein trisections

Bounding the Kirby-Thompson invariant of spun knots

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