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An Exceptional Splitting of Khovanov's Arc Algebras in Characteristic 2

Jesse Cohen

TL;DR

The paper proves a characteristic-2 splitting of Khovanov's arc algebras: over rings with $ ext{char}(R)=2$, the arc algebra satisfies $H_n \cong \tilde{H}_n \otimes A$ with $A=R[x]/(x^2)$ (equivalently $H_n \cong \tilde{H}_n[x]/(x^2)$). It constructs an explicit isomorphism $\\lambda: \tilde{H}_n \otimes A \to H_n$ using a standard basis of labelings and verifies multiplicativity through case analysis that uses $x^2=0$ and saddle cobordisms; a parallel construction extends to bimodules of planar tangles via $\\lambda^L$ and $\\lambda^R$. The authors show that these maps intertwine module structures in characteristic $2$ but need not be bimodule isomorphisms in general. Finally, they prove that no such splitting exists over the integers, using center calculations and known results on invertible central elements, revealing a fundamental integral obstruction. This work connects the modular splitting phenomenon to broader aspects of Khovanov theory and its tangle/bimodule variants, while clarifying the limits of integral lifts.

Abstract

We show that there is an associative algebra $\widetilde{H}_n$ such that, over a base ring $R$ of characteristic 2, Khovanov's arc algebra $H_n$ is isomorphic to the algebra $\widetilde{H}_n[x]/(x^2)$. We also show a similar result for bimodules associated to planar tangles and prove that there is no such isomorphism over $\mathbb{Z}$.

An Exceptional Splitting of Khovanov's Arc Algebras in Characteristic 2

TL;DR

The paper proves a characteristic-2 splitting of Khovanov's arc algebras: over rings with , the arc algebra satisfies with (equivalently ). It constructs an explicit isomorphism using a standard basis of labelings and verifies multiplicativity through case analysis that uses and saddle cobordisms; a parallel construction extends to bimodules of planar tangles via and . The authors show that these maps intertwine module structures in characteristic but need not be bimodule isomorphisms in general. Finally, they prove that no such splitting exists over the integers, using center calculations and known results on invertible central elements, revealing a fundamental integral obstruction. This work connects the modular splitting phenomenon to broader aspects of Khovanov theory and its tangle/bimodule variants, while clarifying the limits of integral lifts.

Abstract

We show that there is an associative algebra such that, over a base ring of characteristic 2, Khovanov's arc algebra is isomorphic to the algebra . We also show a similar result for bimodules associated to planar tangles and prove that there is no such isomorphism over .
Paper Structure (6 sections, 4 theorems, 58 equations, 2 figures)

This paper contains 6 sections, 4 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Background
  4. The Main Theorem
  5. Bimodules of planar tangles
  6. $\mathbb{Z}$-coëfficients

Key Result

Lemma 2.1

Let $\widetilde{m}:\widetilde{H}_n\otimes\widetilde{H}_n\to\widetilde{H}_n$ be the map induced by multiplication on $H_n$. Then $(\widetilde{H}_n,\widetilde{m})$ is a graded associative unital algebra.

Figures (2)

  • Figure 1: The set $\mathfrak{C}_2$ of planar crossingless matchings on $4$ points.
  • Figure 2: An example of $\mathit{Kh}(\Sigma)(\bm{v})$ (left) versus $\widetilde{\mathit{Kh}}(\Sigma)(\bm{v})$ (right) in which the two differ.

Theorems & Definitions (20)

  • Definition 1
  • Lemma 2.1
  • proof
  • Example 1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • proof : Case 1: $s_1=s_2=x$
  • proof : Case 2: $s_1=1$ and $s_2=x$
  • ...and 10 more