[Paper Review] On the spectral sequence from Khovanov homology to Heegaard Floer homology

John A. Baldwin|ArXiv.org|Sep 19, 2008

Geometric and Algebraic Topology22 references15 citations

TL;DR

This paper proves that the $E^k$ terms of the spectral sequence from Khovanov homology to Heegaard Floer homology are link invariants for $k \geq 2$, establishing a new family of graded invariants $\{E^k(L)\}$ that refine classical link invariants. It further shows that Plamenevskaya's transverse invariant lifts to invariants $\psi^k(L)$ in each $E^k(L)$, enabling explicit computation of the spectral sequences for torus knots $T(3,4)$ and $T(3,5)$, which collapse at $E^3$ and $E^4$ respectively.

ABSTRACT

Ozsvath and Szabo show that there is a spectral sequence whose E^2 term is the reduced Khovanov homology of L, and which converges to the Heegaard Floer homology of the (orientation reversed) branched double cover of S^3 along L. We prove that the E^k term of this spectral sequence is an invariant of the link L for all k >= 2. If L is a transverse link in the standard tight contact structure on S^3, then we show that Plamenevskaya's transverse invariant psi(L) gives rise to a transverse invariant, psi^k(L), in the E^k term for each k >= 2. We use this fact to compute each term in the spectral sequences associated to the torus knots T(3,4) and T(3,5).

Motivation & Objective

• To establish that the $E^k$ terms of the Ozsváth–Szabó spectral sequence from Khovanov homology to Heegaard Floer homology are invariants of the link $L$ for $k \geq 2$, resolving a long-standing question about diagram independence.
• To extend Plamenevskaya’s transverse link invariant $\psi(L)$ from Khovanov homology to invariants $\psi^k(L)$ in each $E^k(L)$, thereby enriching the spectral sequence with transverse invariants.
• To compute the full structure of the spectral sequences for the torus knots $T(3,4)$ and $T(3,5)$, including the higher differentials and Poincaré polynomials of $E^k$ terms.
• To investigate the possibility of a quantum grading on $E^k(L)$ and to explore whether higher differentials shift this grading by $2k-2$, as conjectured.
• To lay foundational groundwork for understanding whether link cobordisms induce well-defined maps between $E^k(L)$ terms, generalizing known maps in Khovanov and Heegaard Floer homology.

Proposed method

• Prove that the $E^k$ terms of the spectral sequence are independent of the choice of planar diagram for $L$, using diagrammatic invariance arguments and properties of the spectral sequence construction.
• Construct a transverse invariant $\psi^k(L) \in E^k(L)$ for each $k \geq 2$ by lifting Plamenevskaya’s invariant $\psi(L) \in \kh(L)$ through the spectral sequence.
• Use the fact that $T(3,4)$ and $T(3,5)$ are almost alternating and admit positive braid representatives to constrain the possible higher differentials via grading shifts and non-vanishing cycles.
• Apply the conjectured differential grading shift $D^k$ increasing quantum grading by $2k-2$ and homological grading by $k$, to deduce the structure of $E^k$ terms.
• Compute the Poincaré polynomials of $E^k(T(3,4))$ and $E^k(T(3,5))$ by tracking the survival of generators and the action of $D^k$ across the spectral sequence.
• Use the collapse of the spectral sequence at $E^3$ for $T(3,4)$ and $E^4$ for $T(3,5)$ to determine the final $E^\infty$ terms, which are isomorphic to $\hf(-\Sigma(L))$.

Experimental results

Research questions

• RQ1 Is the $E^k$ term of the spectral sequence from Khovanov homology to Heegaard Floer homology an invariant of the link $L$ for $k \geq 2$, independent of the planar diagram?
• RQ2 Can Plamenevskaya’s transverse invariant $\psi(L)$ in $\kh(L)$ be lifted to a non-trivial invariant $\psi^k(L)$ in each $E^k(L)$ for $k \geq 2$?
• RQ3 What is the structure of the spectral sequence for the torus knots $T(3,4)$ and $T(3,5)$, including the higher differentials and the Poincaré polynomials of $E^k$ terms?
• RQ4 Does the differential $D^k$ on $E^k(L)$ increase the quantum grading by $2k-2$ and the homological grading by $k$, as conjectured?
• RQ5 Do link cobordisms induce well-defined maps between the $E^k$ terms of the spectral sequence, generalizing known maps in Khovanov and Heegaard Floer homology?

Key findings

• The $E^k$ term of the spectral sequence is an invariant of the link $L$ for all $k \geq 2$, resolving a key question about diagram independence in the Ozsváth–Szabó spectral sequence.
• For the torus knot $T(3,4)$, the spectral sequence collapses at $E^3$, with $E^3(T(3,4))$ having Poincaré polynomial $h^0q^6 + h^3q^{12} + h^5q^{16}$, and $\text{rk}\, \hf(-\Sigma(T(3,4))) = 3$.
• For the torus knot $T(3,5)$, the spectral sequence collapses at $E^4$, with $E^4(T(3,5))$ isomorphic to $\mathbb{Z}_2$, and $E^2(T(3,5))$ having Poincaré polynomial $h^0q^8 + h^2q^{12} + h^3q^{14} + h^4q^{14} + h^5q^{18} + h^6q^{18} + h^7q^{20}$.
• The transverse invariant $\psi(L)$ for the positive 3-braid closure of $(xy)^5$ survives as a non-zero cycle $\psi^k(\mathcal{T})$ in each $E^k(T(3,5))$, and generates $E^\infty(T(3,5)) \cong \mathbb{Z}_2$.
• The higher differentials $D^k$ act by increasing homological grading by $k$ and quantum grading by $2k-2$, and this grading shift uniquely determines the spectral sequence structure for $T(3,4)$ and $T(3,5)$.
• The results support the conjecture that $D^k$ increases quantum grading by $2k-2$ and homological grading by $k$, and that $E^k(L)$ admits a well-defined quantum grading for $k \geq 2$.