[Paper Review] A note on Khovanov-Rozansky $sl_2$-homology and ordinary Khovanov homology

Mark C. Hughes|arXiv (Cornell University)|Feb 2, 2013

Topological and Geometric Data Analysis3 references3 citations

TL;DR

This paper establishes a direct isomorphism between Khovanov-Rozansky $sl_2$-homology and ordinary Khovanov homology by resolving long-standing sign ambiguities in the $sl_2$-complex. Using a cube-of-resolutions framework, the author constructs explicit, sign-coherent isomorphisms between the chain complexes of both theories, proving they are isomorphic as bigraded $\mathbb{Q}$-vector spaces with quantum grading reversed. The result confirms a long-standing claim in the literature and clarifies the relationship between the two homology theories.

ABSTRACT

In this note we present an explicit isomorphism between Khovanov-Rozansky $sl_2$-homology and ordinary Khovanov homology. This result was originally stated in Khovanov and Rozansky's paper \cite{KRI}, though the details have yet to appear in the literature. The main missing detail is providing a coherent choice of signs when identifying variables in the $sl_2$-homology. Along with the behavior of the signs and local orientations in the $sl_2$-homology, both theories behave differently when we try to extend their definitions to virtual links, which seemed to suggest that the $sl_2$-homology may instead correspond to a different variant of Khovanov homology. In this paper we describe both theories and prove that they are in fact isomorphic by showing that a coherent choice of signs can be made. In doing so we emphasize the interpretation of the $sl_2$-complex as a cube of resolutions.

Motivation & Objective

• To resolve the missing sign coherence in the isomorphism between Khovanov-Rozansky $sl_2$-homology and ordinary Khovanov homology, a result claimed but not detailed in Khovanov and Rozansky's original paper.
• To clarify why the two theories, despite differing in their treatment of local orientations and signs, are in fact isomorphic over $\mathbb{Q}$.
• To demonstrate that the $sl_2$-homology corresponds directly to ordinary Khovanov homology, not a different variant, by constructing explicit isomorphisms at the level of chain complexes.
• To show that the isomorphism fails to extend to virtual links, highlighting a key distinction in how the two theories behave under generalization.

Proposed method

• Constructs both the ordinary Khovanov complex and the $sl_2$-Khovanov-Rozansky complex as cubes of resolutions, with vertices corresponding to smoothings of link diagrams.
• Assigns quantum grading shifts based on the number of 1-smoothings and crossing signs, using a $\mathbb{Q}$-graded vector space $V = \mathbb{Q}\{1\} \oplus \mathbb{Q}\{x\}$ with $\deg(1) = 1$, $\deg(x) = -1$.
• Defines a sign-coherent isomorphism $\theta_v$ between the vertex spaces of the two cubes by mapping tensor products of $1$ and $x$ to monomials in $\mathbb{Q}[z_1,\dots,z_k]/(z_i^2)$ with appropriate degree shifts.
• Uses matrix factorization techniques and trivalent graph structures to define edge maps in the $sl_2$-complex, ensuring compatibility with the isomorphism via consistent sign conventions.
• Verifies that the isomorphisms commute with the differential maps by checking commutativity of the relevant diagrams using explicit computations involving $\tau(z_i)z_i$ and $z_i^2 = 0$.
• Computes homological and quantum gradings explicitly, showing that the isomorphism preserves homological grading and reverses quantum grading by a sign, confirming the stated isomorphism.

Experimental results

Research questions

• RQ1 Is there a canonical isomorphism between Khovanov-Rozansky $sl_2$-homology and ordinary Khovanov homology, and if so, what is its explicit form?
• RQ2 Why do the two theories appear to differ in behavior when extended to virtual links, and does this indicate a fundamental distinction?
• RQ3 Can a coherent choice of signs be made in the $sl_2$-homology to ensure compatibility with the ordinary Khovanov complex?
• RQ4 Does the $sl_2$-homology truly categorify the Jones polynomial in the same way as ordinary Khovanov homology, or is it a different variant?
• RQ5 What is the precise relationship between the quantum grading in the two theories, and how does it affect the isomorphism?

Key findings

• The paper constructs an explicit isomorphism between the $sl_2$-Khovanov-Rozansky homology and ordinary Khovanov homology as bigraded $\mathbb{Q}$-vector spaces.
• The isomorphism preserves homological grading and reverses quantum grading, so $\mathcal{H}^{i,j}(L) \cong \mathcal{H}^{i,-j}_2(L)$ for all $i,j \in \mathbb{Z}$.
• A coherent choice of signs for the variables $z_i$ in the $sl_2$-complex can be made, resolving a long-standing ambiguity in the literature.
• The isomorphism is compatible with the differential maps in the cube-of-resolutions construction, as verified by explicit computation of edge map commutativity.
• The two theories behave differently when extended to virtual links, indicating that the isomorphism does not extend to the virtual setting.
• The quantum grading shift in the $sl_2$-complex is given by $p_{v'} = -\sum v'_i + n_- - n_+$, which matches the required shift to align with the ordinary Khovanov grading after sign reversal.