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An unoriented skein exact triangle in unoriented link Floer homology

Gheehyun Nahm

TL;DR

This work constructs an unoriented skein theory for link Floer homology by defining band maps that relate three links in a local skein triple. It introduces and analyzes unoriented link Floer homology $\mathrm{HFL'}^{-}$, its hat/reduced variants, and a robust $A_\infty$-framework with twisted complexes to produce exact triangles, including a Heegaard Floer analogue of a $2$-surgery triangle. A key local computation reduces global questions to genus-1 models on the torus, enabling a proof of the unoriented skein exact triangle via triangle counts and positivity arguments. The results connect to Khovanov-type theories (via planar cases) and to Kronheimer–Mrowka’s $I^{\sharp}$-type skein ideas, suggesting spectral sequences from Khovanov homology to unoriented link Floer homology and opening avenues for bordered/grid techniques. The work also clarifies the relationship with Manolescu’s maps, shows exactness in multiple flavors, and lays groundwork for future unoriented link cobordism maps and sign-lit coefficient refinements.

Abstract

We define band maps in unoriented link Floer homology and show that they form an unoriented skein exact triangle. These band maps are similar to the band maps in equivariant Khovanov homology given by the Lee deformation. As a key tool, we use a Heegaard Floer analogue of Bhat's recent 2-surgery exact triangle in instanton Floer homology, which may be of independent interest. Unoriented knot Floer homology corresponds to $I^{\sharp}$ of the knot in our 2-surgery exact triangle.

An unoriented skein exact triangle in unoriented link Floer homology

TL;DR

This work constructs an unoriented skein theory for link Floer homology by defining band maps that relate three links in a local skein triple. It introduces and analyzes unoriented link Floer homology , its hat/reduced variants, and a robust -framework with twisted complexes to produce exact triangles, including a Heegaard Floer analogue of a -surgery triangle. A key local computation reduces global questions to genus-1 models on the torus, enabling a proof of the unoriented skein exact triangle via triangle counts and positivity arguments. The results connect to Khovanov-type theories (via planar cases) and to Kronheimer–Mrowka’s -type skein ideas, suggesting spectral sequences from Khovanov homology to unoriented link Floer homology and opening avenues for bordered/grid techniques. The work also clarifies the relationship with Manolescu’s maps, shows exactness in multiple flavors, and lays groundwork for future unoriented link cobordism maps and sign-lit coefficient refinements.

Abstract

We define band maps in unoriented link Floer homology and show that they form an unoriented skein exact triangle. These band maps are similar to the band maps in equivariant Khovanov homology given by the Lee deformation. As a key tool, we use a Heegaard Floer analogue of Bhat's recent 2-surgery exact triangle in instanton Floer homology, which may be of independent interest. Unoriented knot Floer homology corresponds to of the knot in our 2-surgery exact triangle.
Paper Structure (59 sections, 38 theorems, 112 equations, 30 figures)

This paper contains 59 sections, 38 theorems, 112 equations, 30 figures.

Table of Contents

  1. Introduction
  2. Unoriented link Floer homology
  3. An unoriented skein exact triangle
  4. Unlinks and the Hopf link
  5. Trefoil knots
  6. Comparing with Manolescu's maps MR2350128
  7. A word on the proof
  8. Motivation
  9. Future directions
  10. Organization
  11. Acknowledgements
  12. Preliminaries
  13. Heegaard Floer homology and local systems
  14. Heegaard data for unoriented link Floer homology
  15. Weak admissibility and positivity
  16. ...and 44 more sections

Key Result

Theorem 1.2

Let $L_{a},L_{b},L_{c}\subset Y$ be an unoriented skein triple. Then there exists an $\mathbb{F}\llbracket U\rrbracket$-linear exact triangle

Figures (30)

  • Figure 1.1: A local diagram for an unoriented skein triple
  • Figure 1.2: A local diagram for an unoriented skein triple together with the band maps
  • Figure 1.3: A non-orientable band between two unknots for Manolescu's exact triangle, and a Heegaard diagram for it
  • Figure 1.4: A non-orientable band between two unknots for our exact triangle, and a Heegaard diagram for it
  • Figure 1.5: We reduce the proof of Theorem \ref{['thm:unoriented-exact']} to a local computation, Theorem \ref{['thm:sym2-t2']}.
  • ...and 25 more figures

Theorems & Definitions (161)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 151 more