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Regularly slice implies once-stably decomposably slice

Joseph Breen

Abstract

We investigate the relationship between regular and decomposable Lagrangian cobordisms in $4$-dimensional symplectizations. First, we show that regular sliceness implies once-stably decomposable sliceness, and offer a stabilization-free strategy. On the other hand, we show that satelliting preserves regularity of concordance, suggesting that regularity and decomposability are distinct in general. Among other results, we compare the symplectic and smooth slice-ribbon conjectures and construct decomposably slice knots that may not be strongly decomposably slice.

Regularly slice implies once-stably decomposably slice

Abstract

We investigate the relationship between regular and decomposable Lagrangian cobordisms in -dimensional symplectizations. First, we show that regular sliceness implies once-stably decomposable sliceness, and offer a stabilization-free strategy. On the other hand, we show that satelliting preserves regularity of concordance, suggesting that regularity and decomposability are distinct in general. Among other results, we compare the symplectic and smooth slice-ribbon conjectures and construct decomposably slice knots that may not be strongly decomposably slice.

Paper Structure

This paper contains 30 sections, 25 theorems, 17 equations, 37 figures.

Table of Contents

  1. Introduction
  2. Exact, regular, and decomposable cobordisms
  3. Main results
  4. Regularly slice implies once-stably decomposably slice
  5. Diagrammatic presentation of regular cobordisms
  6. Regularity and satelliting
  7. Regularity and normal rulings
  8. Organization
  9. Gradations of the slice-ribbon conjecture
  10. Smooth gradations
  11. Symplectic gradations
  12. Background
  13. Exact and decomposable cobordisms
  14. Weinstein topology
  15. Cotangent bundles
  16. ...and 15 more sections

Key Result

Theorem 1.4

Every regularly slice Legendrian knot is once-stably (strongly) decomposably slice. That is, if $U \prec_{\mathrm{reg}} \Lambda$, then $S_{\pm}(U) \prec_{\mathrm{dec}} S_{\pm}(\Lambda)$.

Figures (37)

  • Figure 1: The decomposable moves, unknot birth and (ambient) Legendrian surgery.
  • Figure 2: A schematic picture of \ref{['thm:main3']}. On the left is a regular slice disk, given as a concatenation of the standard filling of $U$ with a regular concordance. On the right is a decomposable concordance between the once-stabilized knots. The box is a placeholder for a Legendrian tangle.
  • Figure 3: A regular disk filling of $\Lambda_+$, a Legendrian $\overline{9_{46}}$.
  • Figure 4: A regular (saddle) cobordism as described by \ref{['thm:cob_diag']} from the trefoil $\Lambda_-$ to some two-component Legendrian link $\Lambda_+$. Here $n=k=1$.
  • Figure 5: The diagrammatic presentation of a genus $g\geq 1$ filling on the left and continuing on the top right; an example of a regular concordance on the lower right. In fact, the concordance is from the trefoil to the Whitehead double of $\overline{9_{46}}$ as considered by cornwell2016concordance; see the discussion in \ref{['subsec:sat']}.
  • ...and 32 more figures

Theorems & Definitions (60)

  • Remark 1.1
  • Theorem 1.4
  • Remark 1.7
  • Theorem 1.8: Regular cobordism diagrams
  • Corollary 1.9: Regular filling diagrams
  • Corollary 1.10: Regular concordance diagrams
  • Theorem 1.11
  • Corollary 1.12
  • Corollary 1.14
  • Theorem 1.15
  • ...and 50 more