A Colored Khovanov Homotopy Type And Its Tail For B-Adequate Links

TL;DR

This paper introduces a new colored Khovanov homotopy type for links and spin networks, demonstrating stabilization properties for B-adequate links as the coloring increases, extending the understanding of tail behaviors in quantum invariants.

Contribution

It defines a novel colored Khovanov homotopy type for links and spin networks, and establishes stabilization results for B-adequate links as the color parameter grows.

Findings

Stabilization of homotopy types for B-adequate links as color increases

Generalization of tail behavior of colored Jones polynomial

Simplified stabilization for colored unknot

Abstract

We define a Khovanov homotopy type for $sl_2(\mathbb{C})$ colored links and quantum spin networks and derive some of its basic properties. In the case of $n$-colored B-adequate links, we show a stabilization of the homotopy types as the coloring $n\rightarrow\infty$, generalizing the tail behavior of the colored Jones polynomial. Finally, we also provide an alternative, simpler stabilization in the case of the colored unknot.