Khovanov complexes of rational tangles

TL;DR

This paper demonstrates a simplified, computationally efficient representation of Khovanov complexes for rational tangles, revealing a connection to the reduced Burau representation of braid groups.

Contribution

It introduces a minimal, zig-zag structured complex for rational tangles and links it to the reduced Burau representation, enabling quick calculations.

Findings

Minimal complexes have a zig-zag backbone

The complexes can be computed efficiently using an inductive algorithm

Bigradings relate to the reduced Burau representation

Abstract

We show that the Khovanov complex of a rational tangle has a very simple representative whose backbone of non-zero morphisms forms a zig-zag. Furthermore, this minimal complex can be computed quickly by an inductive algorithm. (For example, we calculate $Kh(8_2)$ by hand.) We find that the bigradings of the subobjects in these minimal complexes can be described by matrix actions, which after a change of basis is the reduced Burau representation of $B_3$.