[Paper Review] Wall-Crossing Morphisms in Khovanov-Rozansky Homology

Nadya Shirokova, Ben Webster|ArXiv.org|Jun 11, 2007

Geometric and Algebraic Topology6 references3 citations

TL;DR

This paper constructs a wall-crossing morphism for Khovanov-Rozansky homology, enabling a categorification of the Vassiliev derivative for HOMFLY polynomial and $\mathfrak{sl}_N$ quantum invariants. By defining commuting wall-crossing maps between KR homologies of links differing by a crossing change, the authors extend KR homology to singular knots via an iterated cone construction, yielding a finite-dimensional triply graded homology theory whose Euler characteristic matches the Vassiliev derivative of the HOMFLY polynomial.

ABSTRACT

We define a wall-crossing morphism for Khovanov-Rozansky homology; that is, a map between the KR homology of knots related by a crossing change. Using this map, we extend KR homology to an invariant of singular knots categorifying the Vasilliev derivative of the HOMFLY polynomial, and of $\mathfrak{sl}_n$ quantum invariants.

Motivation & Objective

• To define a wall-crossing morphism in Khovanov-Rozansky homology that categorifies the Vassiliev derivative of knot invariants.
• To extend KR homology to singular knots by constructing a consistent, commuting system of wall-crossing maps for links differing by a single crossing change.
• To define a triply graded homology theory for singular links whose Euler characteristic is the Vassiliev derivative of the HOMFLY polynomial.
• To ensure the wall-crossing maps commute despite being defined only up to scalar, enabling consistent extension to higher-order singularities.
• To generalize the framework of categorified Vassiliev theory to include Khovanov-Rozansky homology, building on earlier work on finite-type invariants.

Proposed method

• Define a wall-crossing morphism $\mathcal{W}_w: \mathcal{KR}_N(K_-) \to \mathcal{KR}_N(K_+)$ for each codimension-1 wall in the discriminant of the space of knots.
• Construct the wall-crossing map via the Yoneda product in $\mathrm{Ext}^1$ groups, using a specific exact sequence involving tensor products of polynomial algebras and braid group functors.
• Realize the wall-crossing map as the image of a unique (up to scalar) projection in $\mathrm{Ext}^1(F(\sigma_i^{-1}), F(\sigma_i))$, ensuring functoriality and compatibility with braid group actions.
• Form the total complex $\mathcal{VKR}_N(K)$ as an iterated cone over all wall-crossing maps for a singular link $K$, with grading inherited from the KR homology of resolutions.
• Use spectral sequences and functoriality of tensor products to verify invariance under Markov moves, ensuring the homology is a well-defined link invariant.
• Prove finiteness of homology via the spectral sequence of a double complex whose components are finite-dimensional KR homologies of resolutions.

Experimental results

Research questions

• RQ1 Can a wall-crossing morphism be defined for Khovanov-Rozansky homology that categorifies the Vassiliev derivative of knot invariants?
• RQ2 How can KR homology be extended to singular knots in a way that categorifies the Vassiliev derivative of the HOMFLY polynomial?
• RQ3 Do the wall-crossing maps for adjacent singularities commute, allowing consistent extension to higher-order singular knots?
• RQ4 Is the resulting homology theory finite-dimensional for any singular knot?
• RQ5 Can this construction be applied to the triply graded KR homology, where projective functoriality is not yet established?

Key findings

• A wall-crossing morphism $\mathcal{W}_w: \mathcal{KR}_N(K_-) \to \mathcal{KR}_N(K_+)$ is explicitly constructed for all $N = 1, \dots, \infty$, defined via $\mathrm{Ext}^1$ classes in a derived category of bimodules.
• The wall-crossing maps for adjacent singularities commute under the Yoneda product, ensuring consistency in the iterated cone construction over multiple singularities.
• The total complex $\mathcal{VKR}_N(K)$ for a singular link $K$ is a well-defined, finite-dimensional homology theory, with homology finite-dimensional due to the spectral sequence of the double complex of resolutions.
• The Euler characteristic of $\mathcal{VKR}_\infty(K)$ is the Vassiliev derivative of the HOMFLY polynomial, and for $N < \infty$, it categorifies the Vassiliev derivative of the $\mathfrak{sl}_N$ quantum invariant.
• The construction applies to the triply graded KR homology, even though projective functoriality is not yet known in that setting.
• The wall-crossing maps are defined only up to scalar, but consistent scalar choices ensure that the iterated cone is well-defined and invariant under isotopy.