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Weinstein presentations for high-dimensional antisurgery

Ipsita Datta, Oleg Lazarev, Chindu Mohanakumar, Angela Wu

TL;DR

This work addresses the problem of explicitly describing Weinstein presentations for high-dimensional subdomains obtained by antisurgery along regular Lagrangian disks. It introduces the boat move, a higher-dimensional Legendrian isotopy, together with $D^k$-suspensions to realize antisurgery in front projections, enabling explicit handlebody decompositions for $X\setminus L$. The main result shows that for $X$ built from $T^*D^n$ with $n$-handles, the complement by a regular disk $L$ admits a presentation where each attaching sphere is modified by a corresponding $(n,n-k)$-boat move on Reeb chords of local index $k$, followed by cusp summations; this also yields explicit $P$-loose Legendrians and $P$-loose domains. The paper provides explicit exotic Weinstein subdomains as applications and discusses open problems, linking geometric antisurgery to categorical invariants and localization in wrapped Fukaya categories.

Abstract

In this paper, we give an algorithm for describing the Weinstein presentation of Weinstein subdomains obtained by carving out regular Lagrangians. Our work generalizes previous work in dimension three and requires a novel Legendrian isotopy move (the ``boat move") that changes the local index of Reeb chords in a front projection. As applications, we describe presentations for certain exotic Weinstein subdomains and give explicit descriptions of $P$-loose Legendrians.

Weinstein presentations for high-dimensional antisurgery

TL;DR

This work addresses the problem of explicitly describing Weinstein presentations for high-dimensional subdomains obtained by antisurgery along regular Lagrangian disks. It introduces the boat move, a higher-dimensional Legendrian isotopy, together with -suspensions to realize antisurgery in front projections, enabling explicit handlebody decompositions for . The main result shows that for built from with -handles, the complement by a regular disk admits a presentation where each attaching sphere is modified by a corresponding -boat move on Reeb chords of local index , followed by cusp summations; this also yields explicit -loose Legendrians and -loose domains. The paper provides explicit exotic Weinstein subdomains as applications and discusses open problems, linking geometric antisurgery to categorical invariants and localization in wrapped Fukaya categories.

Abstract

In this paper, we give an algorithm for describing the Weinstein presentation of Weinstein subdomains obtained by carving out regular Lagrangians. Our work generalizes previous work in dimension three and requires a novel Legendrian isotopy move (the ``boat move") that changes the local index of Reeb chords in a front projection. As applications, we describe presentations for certain exotic Weinstein subdomains and give explicit descriptions of -loose Legendrians.
Paper Structure (16 sections, 10 theorems, 36 equations, 21 figures)

This paper contains 16 sections, 10 theorems, 36 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Structure of the paper
  4. Acknowledgements
  5. Background
  6. Contact surgery and Weinstein manifolds
  7. Legendrian moves
  8. Loose and $P$-loose Legendrians, flexible and $P$-flexible domains
  9. Antisurgery construction of $P$-loose Legendrian unknots
  10. Constructions of Higher Dimensional Legendrian Isotopy Moves
  11. Suspensions of Legendrian isotopies
  12. Boat move
  13. Weinstein presentations of subdomains
  14. Proof of Main Theorem \ref{['thm: general_antisurgery_intro']}
  15. Explicit examples
  16. ...and 1 more sections

Key Result

Proposition 1.1

Given a Legendrian isotopy $\psi: D^{n-k} \times [0,1]_t \to \mathbb{R}^{2(n-k) + 1}$ which is the identity near $\partial D^{n-k}$ and t-independent near $\partial [0,1]$, its $D^k$-suspension, $\Sigma_{D^k} \{\psi\}$ is a Legendrian in $\mathbb{R}^{2n+1}$, and is Legendrian isotopic to $D^k \times

Figures (21)

  • Figure 1: The green Reeb chords are bounded by the flying saucer and go from the black $\Lambda_i$ to the red $\Lambda_\emptyset$; these have critical points on $\Lambda_i$ with local index $1, 0, 1$ for the height difference Morse function from $\Lambda_i$ to $\Lambda_\emptyset$. We will apply $(1,0), (1,1), (1,0)$ boat moves at these critical points respectively, which correspond to doing a Reidemeister 1 move at the two index 1 critical points and nothing at the middle index 0 critical point. The purple Reeb chord is not bounded by the flying saucer. The pink Reeb chord is bounded by the flying saucer but does not go from $\Lambda_i$ to $\Lambda_\emptyset$.
  • Figure 2: A cartoon depicting a low-dimensional slice of a $P$-loose Legendrian which cuts through the three boat move cusp connect sums, represented by the pink regions. Away from a neighbourhood of this slice, the black Legendrians may be linked in the yellow region but are parallel outside of it.
  • Figure 3: The Legendrian Reidemeister moves.
  • Figure 4: A Legendrian Reidemeister 1 and isotopy for a knot (top) and a surface (bottom).
  • Figure 5: Handle slides over (-1) and (+1) Legendrians (in blue and red respectively) in 3 and 5 dimensions.
  • ...and 16 more figures

Theorems & Definitions (39)

  • Proposition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Theorem 2.1: See Swiatowski_1992
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • ...and 29 more