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Spectral sequences in unoriented link Floer homology

Gheehyun Nahm

TL;DR

The paper proves Rasmussen's conjecture over the field $\\mathbb{F}=\\mathbb{Z}/2$ by constructing a spectral sequence from the reduced Khovanov homology of the mirror, $\\widetilde{Kh}(m(K))$, to knot Floer homology $\\widehat{HFK}(S^{3},K)$ for knots in $S^{3}$, and extends this framework to links in arbitrary 3-manifolds via unoriented link Floer homology. It develops an algebraic and geometric apparatus built on pair-pointed Heegaard diagrams, $A_\infty$-categories, and band/swap maps to iteratively apply a modified unoriented skein triangle, producing a cube of resolutions whose $E_{1}$ page recovers Khovanov data and whose $E_{\infty}$ page yields $HFL'$. The work establishes Alexander $\mathbb{Z}/2$-splittings, relative gradings, and natural basepoint actions that intertwine with $A_\infty$-products, enabling precise comparisons between Khovanov and unoriented link Floer homology. It provides explicit computations for simple links (e.g., Hopf link, trefoil) and outlines future directions, including signs, integer coefficients, bordered/grid approaches, and connections to Dowlin’s oriented spectral sequence. Overall, the paper delivers a robust, coefficient-two spectral-sequence framework linking Khovanov and unoriented link Floer homologies with potential for new concordance and genus bounds.

Abstract

In a previous work, we defined an unoriented skein exact triangle in unoriented link Floer homology. In this paper, we iterate a modified version of this exact triangle and obtain a spectral sequence from various versions of Khovanov homology to various versions of unoriented link Floer homology, over the field with two elements. In particular, for knots in $S^{3}$, we obtain a spectral sequence from the reduced Khovanov homology of the mirror of the knot to the knot Floer homology of the knot.

Spectral sequences in unoriented link Floer homology

TL;DR

The paper proves Rasmussen's conjecture over the field by constructing a spectral sequence from the reduced Khovanov homology of the mirror, , to knot Floer homology for knots in , and extends this framework to links in arbitrary 3-manifolds via unoriented link Floer homology. It develops an algebraic and geometric apparatus built on pair-pointed Heegaard diagrams, -categories, and band/swap maps to iteratively apply a modified unoriented skein triangle, producing a cube of resolutions whose page recovers Khovanov data and whose page yields . The work establishes Alexander -splittings, relative gradings, and natural basepoint actions that intertwine with -products, enabling precise comparisons between Khovanov and unoriented link Floer homology. It provides explicit computations for simple links (e.g., Hopf link, trefoil) and outlines future directions, including signs, integer coefficients, bordered/grid approaches, and connections to Dowlin’s oriented spectral sequence. Overall, the paper delivers a robust, coefficient-two spectral-sequence framework linking Khovanov and unoriented link Floer homologies with potential for new concordance and genus bounds.

Abstract

In a previous work, we defined an unoriented skein exact triangle in unoriented link Floer homology. In this paper, we iterate a modified version of this exact triangle and obtain a spectral sequence from various versions of Khovanov homology to various versions of unoriented link Floer homology, over the field with two elements. In particular, for knots in , we obtain a spectral sequence from the reduced Khovanov homology of the mirror of the knot to the knot Floer homology of the knot.
Paper Structure (86 sections, 53 theorems, 181 equations, 48 figures, 1 table)

This paper contains 86 sections, 53 theorems, 181 equations, 48 figures, 1 table.

Table of Contents

  1. Introduction
  2. Organization
  3. Future directions
  4. Conventions and notations
  5. Acknowledgements
  6. Unoriented link Floer homology
  7. Pair-pointed Heegaard diagrams
  8. Unoriented links and ${\rm Spin}^{c}$-structures
  9. Strong admissibility and unoriented link Floer homology
  10. The Ainf-categories that we work in
  11. An Alexander $\mathbb{Z}/2$-splitting
  12. Balled links and unoriented link Floer homology
  13. More Heegaard Floer preliminaries
  14. The Phi action
  15. Relative homology actions
  16. ...and 71 more sections

Key Result

Theorem 1.1

Let $K$ be a knot in $S^{3}$. Then, there exists a spectral sequence over the field $\mathbb{F}=\mathbb{Z}/2$ with two elements, where $\widetilde{Kh}(m(K))$ is the reduced Khovanov homology of the mirror of the knot.

Figures (48)

  • Figure 3.1: Connected summing region
  • Figure 3.2: These annuli, viewed as domains in $D(ax,bx)$, always have an odd number of holomorphic representatives.
  • Figure 3.3: Left: a local Heegaard diagram for the non-orientable band map. Right: a local Heegaard diagram for the orientable band maps. The basepoint pairs $(z,w)$ and $(z_{i},w_{i})$ are link basepoint pairs for all the attaching curves, and the forbidding arcs are omitted.
  • Figure 4.1: Our convention for Khovanov homology
  • Figure 5.1: A local diagram for an unoriented skein triple in $D^{3}$ and bands between them
  • ...and 43 more figures

Theorems & Definitions (155)

  • Theorem 1.1
  • Theorem 1.2: Corollary \ref{['cor:khovanov-main']}
  • Remark 1.3
  • Corollary 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • ...and 145 more