Thin links and Conway spheres

Abstract: When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal $\delta$-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.

• The paper introduces a novel characterization of tangle thinness by decomposing links using Conway spheres.
• It establishes intrinsic geometric ties between knot homologies and 3-manifold topology without solely relying on classical invariants.
• The work extends quasi-alternating link theorems by providing new insights into tangle homologies and their relation to taut foliations.

Thin Links and Conway Spheres: A Summary

The paper "Thin Links and Conway Spheres" by Kotelskiy, Watson, and Zibrowius, addresses an intricate problem within the framework of low-dimensional topology, specifically focusing on knot theory and its associated homological invariants. The crux of the paper rests on characterizing thin links, a term used to describe links whose homological invariants, such as Khovanov homology and Heegaard Floer homology, exhibit minimal support along a single diagonal grading. In addition, the paper introduces a novel approach to understanding these structures through tangle decompositions along Conway spheres.

The concept of thinness is grounded in the properties of Khovanov homology and knot Floer homology, known to thin out in alternating links strictly along a $\delta$-grading. Instead of relying solely on homological invariants, the authors propose a characterization through relative versions of thinness for tangles. This aligns with Fox's classical question about characterizing alternating knots, working instead towards a comprehensive depiction of thin links, which defies dependence on specific homological theories.

Main Contributions

1. Characterization of Tangle Thinness: The paper offers a new characterization of thinness in tangles via tangle decompositions with Conway spheres. The results intimate a parallel to the L-space gluing theorem for three-manifolds with torus boundaries, suggesting a topological characterization without express reference to homological invariants.
2. Geometric and Topological Interactions: By leveraging the connection between knot homologies and tangle decompositions, the authors propose that thinness has an intrinsic geometry. The authors speculatively extend these observations to imply relations between thin links and three-manifold topology, especially concerning L-spaces and taut foliations.
3. Implications for Taut Foliations and Left-orderable Fundamental Groups: The paper touches on the burgeoning conjectural framework that relates the existence of taut foliations, the non-orderability of fundamental groups, and non-L-space characteristics, supplementing these with tangle-thinness results.
4. Extending Quasi-alternating Link Theorems: Previous results, such as those indicating quasi-alternating links exhibit thinness, are extended by showing that tangles exhibiting these characteristics can decompose without directly referencing the homological setting.
5. Connection to Tangle Homologies: In support of the theoretical development, the authors present a new invariant, generalized from previously developed Heegaard Floer invariants and evaluated against Conway spheres.

Analytical Framework and Results

• The authors develop an analytical framework using immersed curves on four-punctured spheres to capture the essence of thin links via parametrized tangles.
• They prove that for a certain class of tangles—rational tangles—characterization is simple yet elegant, with a direct link to the frequency and positioning of $\delta$-Curvatures in the surface intersection.
• They extend the discussion to homogeneous $\delta$-graded vector spaces, providing a topological interpretation for thin and A-links, further elucidated with deep combinatorial discourse.

The paper is heavily mathematical, with rigorous proofs culminating in a noteworthy gluing theorem that unifies and extends the understanding of thin links as part of the broader study of low-dimensional topology. Through implicative reasoning, the authors leave room for future exploration into the potential connections amongst homological assignations, especially in the context of branched covers and other manifold interactions. This comprehensive mathematical treatise promises a substantial contribution to knot theory and three-manifold topology, enriching and potentially revising classical perspectives within the field.