[Paper Review] Odd Khovanov homology for tangles

Grégoire Naisse, Krzysztof K. Putyra|arXiv (Cornell University)|Mar 31, 2020

Algebraic structures and combinatorial models24 references4 citations

TL;DR

This paper extends odd Khovanov homology to tangles using quasi-associative algebras and bimodules graded over a category with a 3-cocycle, resolving non-associativity in odd arc algebras. It constructs a G-graded 2-functor that recovers the odd Khovanov tangle invariant via a covering version of the level 2 cyclotomic half 2-Kac–Moody algebra, proving compatibility with Vaz's construction and establishing a gluing property for tangle invariants.

ABSTRACT

We extend the covering of even and odd Khovanov link homology to tangles, using arc algebras. For this, we develop the theory of quasi-associative algebras and bimodules graded over a category with a 3-cocycle. Furthermore, we show that a covering version of a level 2 cyclotomic half 2-Kac--Moody algebra acts on the bicategory of quasi-associative bimodules over the covering arc algebras, relating our work to a construction of Vaz.

Motivation & Objective

• To resolve the non-associativity of odd arc algebras, which obstructs defining tangle invariants via standard bimodule tensor products.
• To develop a theory of quasi-associative algebras and bimodules graded over a category equipped with a 3-cocycle (associator), enabling consistent tangle gluing.
• To construct a grading shift functor system for quasi-associative bimodules, generalizing supercategory parity shifts.
• To establish a G-graded 2-functor from a chronological cobordism 2-category to a bicategory of quasi-associative bimodules, yielding a tangle invariant.
• To show that this construction recovers odd Khovanov homology for links and agrees with Vaz's odd tangle invariant via a 2-action of the cyclotomic half 2-Kac–Moody algebra.

Proposed method

• Introduce a G-graded category of tangles with morphisms labeled by flat tangles and Z×Z gradings, enabling consistent grading of arc algebras.
• Define quasi-associative algebras and bimodules over a category with a 3-cocycle, generalizing Albuquerque–Majid’s framework to handle non-associative structures.
• Construct a shifting-2-system over the grading category to model grading shift functors, generalizing supercategory shift functors.
• Define a G-graded 2-functor FL from the 2-category of chronological cobordisms to the bicategory of quasi-associative bimodules over covering arc algebras.
• Prove that FL is a G-graded 2-functor satisfying coherence conditions (unitor and associator diagrams), ensuring compatibility with tangle gluing.
• Specialize the covering arc algebra over R = Z[X,Y,Z±1]/(X²=Y²=0) to X=Z=1, Y=−1 to recover odd Khovanov homology and show agreement with Vaz’s tangle invariant.

Experimental results

Research questions

• RQ1 How can odd Khovanov homology be extended from links to tangles, given the non-associativity of odd arc algebras?
• RQ2 What algebraic structure generalizes associativity in the presence of a 3-cocycle, allowing consistent tangle gluing in the odd setting?
• RQ3 How can grading shift functors be defined for quasi-associative bimodules to model tangle invariants with non-trivial grading?
• RQ4 Does the 2-action of the cyclotomic half 2-Kac–Moody algebra on quasi-associative bimodules recover Vaz’s odd tangle invariant?
• RQ5 Is the G-graded 2-functor FL fully faithful, implying equivalence between the new tangle invariant and Vaz’s construction?

Key findings

• The paper constructs a covering version of odd Khovanov homology for tangles using quasi-associative bimodules over a grading category with a 3-cocycle, resolving the non-associativity issue in odd arc algebras.
• A G-graded 2-functor FL is defined from the 2-category of chronological cobordisms to the bicategory of quasi-associative bimodules, satisfying all coherence conditions for a 2-functor.
• The construction satisfies the gluing property: Kh(T′) ⊗Hn Kh(T) ≅ Kh(T′T), generalizing the even case to the odd setting.
• Specializing the covering arc algebra with X=Z=1, Y=−1 yields a tangle invariant that agrees with odd Khovanov homology for links.
• The 2-action of the cyclotomic odd half 2-Kac–Moody algebra on the bicategory of quasi-associative bimodules reproduces the complex of 1-morphisms from Vaz’s construction.
• The authors conjecture that the 2-functor FL is fully faithful, implying that Vaz’s tangle invariant is equivalent to the new odd Khovanov tangle invariant.