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Thin links and Conway spheres

Artem Kotelskiy, Liam Watson, Claudius Zibrowius

Abstract

When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal $δ$-grading. This leads to the broader class of thin links that one would like to characterize without reference to the invariant in question. We provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain immersed curve invariants for Conway tangles, namely the Heegaard Floer invariant $\operatorname{HFT}$ and the Khovanov invariant $\operatorname{\widetilde{Kh}}$ that were developed by the authors in previous works.

Thin links and Conway spheres

Abstract

When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal -grading. This leads to the broader class of thin links that one would like to characterize without reference to the invariant in question. We provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain immersed curve invariants for Conway tangles, namely the Heegaard Floer invariant and the Khovanov invariant that were developed by the authors in previous works.

Paper Structure

This paper contains 29 sections, 72 theorems, 97 equations, 27 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Abstracting the main argument
  3. Combinatorics of slopes and lines
  4. Characterizing thin pairs of line sets
  5. The tangle invariant HFT
  6. The definition of HFT
  7. A gluing theorem for HFT
  8. The geography problem for HFT
  9. Heegaard Floer thin fillings
  10. The δ-grading of curves in the covering space
  11. Linear curves
  12. Heegaard Floer thin fillings
  13. The tangle invariant Khr
  14. The definition of Khr
  15. A gluing theorem for Khr
  16. ...and 14 more sections

Key Result

Proposition \oldthetheorem

Given a link homology theory $\mathop{\mathrm{\mathbf{H}_\ast}}\nolimits$ with full support, a link is thin if and only if it is an A-link.

Figures (27)

  • Figure 1: Two Conway tangle decompositions defining the link $T_1\cup T_2$. The tangle $T_2$ is the result of rotating $T_2$ around the vertical axis. By rotating the entire link on the right-hand side around the vertical axis, we can see that $T_1\cup T_2=T_2\cup T_1$.
  • Figure 2: The paper's sections and their dependencies. Dashed arrows indicate dependencies that need only statements of results and not the machinery that arise in the proofs, so that the sections in each column may be read in isolation.
  • Figure 3: The thin interval relative to an increasing sequence of slopes $(s_1,s_2,s_3,\ldots,s_n)$
  • Figure 4: An illustration of Example \ref{['exa:ExceptionalExample']} for the case $\Delta_c=0=\Delta_d$
  • Figure 5: A simple non-rational tangle and its Heegaard Floer tangle invariant
  • ...and 22 more figures

Theorems & Definitions (166)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem: L-space Gluing Theorem
  • Theorem \oldthetheorem: Characterization of A-link filling spaces
  • Theorem \oldthetheorem: Characterization of thin filling spaces
  • Proposition \oldthetheorem
  • ...and 156 more