[Paper Review] Calculating Bar-Natan's characteristic two Khovanov homology

Paul Turner|ArXiv.org|Nov 10, 2004

Geometric and Algebraic Topology2 references4 citations

TL;DR

This paper develops computational tools for Bar-Natan's characteristic two Khovanov homology, introducing a spectral sequence framework to compute both the filtered and bi-graded theories. The key contribution is identifying the $E_2$-page of the spectral sequence in terms of secondary homology groups derived from an endomorphism on ${\mathbb{F}}_2$-Khovanov homology, enabling explicit computation of the filtered theory via linking numbers and providing a stable limit isomorphic to the singly-graded theory.

ABSTRACT

We investigate Bar-Natan's characteristic two Khovanov link homology theory studying both the filtered and bi-graded theories. The filtered theory is computed explicitly and the bi-graded theory analysed by setting up a family of spectral sequences. The E_2-pages can be described in terms of groups arising from the action of a certain endomorphism on mod 2 Khovanov homology. Some simple consequences are discussed.

Motivation & Objective

• To compute Bar-Natan's characteristic two Khovanov homology, both in its filtered and bi-graded forms, for links over ${\mathbb{F}}_2$.
• To establish a spectral sequence framework that relates the $E_1$-page of ${\mathbb{F}}_2$-Khovanov homology to the filtered Bar-Natan homology.
• To define and analyze an endomorphism $\beta_*$ on ${\mathbb{F}}_2$-Khovanov homology of bi-degree $(1,2)$, whose homology yields secondary groups used in the $E_2$-page computation.
• To prove that the graded Bar-Natan theory stabilizes in the $q$-grading, with the stable limit isomorphic to the filtered theory.
• To provide explicit formulas for the dimensions of the filtered homology in terms of linking numbers of link components.

Proposed method

• Define an endomorphism $\beta_*$ on ${\mathbb{F}}_2$-Khovanov homology of bi-degree $(1,2)$, which satisfies $\beta_*^2 = 0$, allowing the construction of secondary homology groups $\mathbb{K}^{*,*}(L)$.
• Construct a spectral sequence with $E_1$-page isomorphic to ${\mathbb{F}}_2$-Khovanov homology, converging to the filtered Bar-Natan homology $\mathrm{BN}^*(L)'$.
• Use the TQFT associated to the Frobenius algebra ${\mathbb{F}}_2[u]\{1,x\}$ with modified multiplication and comultiplication to define the Bar-Natan complex $\mathcal{C}^{*,*}(D)$ and its differential $d$.
• Establish a quasi-isomorphism between a subcomplex of the Bar-Natan complex and the original Khovanov complex, enabling the transfer of structure and computation.
• Analyze the action of the endomorphism $\beta_*$ on the $q$-graded components of the homology, leading to the $E_2$-page description in terms of $\mathbb{K}^{*,*}(L)$.
• Prove that the spectral sequence for the graded theory stabilizes in the $q$-grading, and that the stable limit is isomorphic to the filtered theory.

Experimental results

Research questions

• RQ1 How can the filtered Bar-Natan homology $\mathrm{BN}^*(L)'$ be computed explicitly for links over ${\mathbb{F}}_2$?
• RQ2 What is the structure of the $E_2$-page of the spectral sequence converging to $\mathrm{BN}^*(L)'$?
• RQ3 How does the endomorphism $\beta_*$ on ${\mathbb{F}}_2$-Khovanov homology relate to the spectral sequence structure of Bar-Natan homology?
• RQ4 Does the graded Bar-Natan homology stabilize in the $q$-grading, and if so, is the stable limit isomorphic to the filtered theory?
• RQ5 Can the dimension of $\mathrm{BN}^*(L)'$ be expressed in terms of linking numbers of the link components?

Key findings

• The dimension of the filtered Bar-Natan homology $\mathrm{BN}^*(L)'$ is $2^k$, where $k$ is the number of components in the link $L$.
• The dimension of $\mathrm{BN}^i(L)'$ is equal to the number of subsets $E \subset \{1, \dots, k\}$ such that $2 \sum_{l \in E, m \notin E} \mathrm{lk}(L_l, L_m) = i$, explicitly linking the homology to linking numbers.
• The $E_2$-page of the spectral sequence converging to $\mathrm{BN}^*(L)'$ is isomorphic to the secondary homology groups $\mathbb{K}^{*,*}(L)$, defined as the homology of $\mathrm{KH}^{*,*}_{\mathbb{F}_2}(L)$ under the differential $\beta_*$.
• The spectral sequence for the graded Bar-Natan theory stabilizes in the $q$-grading, and the stable limit is isomorphic to the filtered Bar-Natan homology $\mathrm{BN}^*(L)'$.
• The spectral sequence for the graded theory coincides with the filtered theory's spectral sequence in the stable $q$-range, confirming consistency across theories.
• The quasi-isomorphism $\phi$ between complexes commutes with the endomorphism $\beta$ up to boundaries, ensuring compatibility of the spectral sequence structure.