On the universal sl_2 invariant of boundary bottom tangles

TL;DR

This paper proves an improved version of Habiro's conjecture regarding the universal sl_2 invariant of boundary bottom tangles, showing it takes values in specific subalgebras and applying this to divisibility properties of colored Jones polynomials.

Contribution

The paper establishes a refined version of Habiro's conjecture, demonstrating the algebraic nature of the universal sl_2 invariant for boundary bottom tangles and its implications.

Findings

Universal sl_2 invariant lies in certain subalgebras of U_h(sl_2) tensor powers.

Proves a divisibility property of the colored Jones polynomial for boundary links.

Provides a deeper understanding of the algebraic structure underlying boundary bottom tangles.

Abstract

The universal sl_2 invariant of bottom tangles has a universality property for the colored Jones polynomial of links. Habiro conjectured that the universal sl_2 invariant of boundary bottom tangles takes values in certain subalgebras of the completed tensor powers of the quantized enveloping algebra U_h(sl_2) of the Lie algebra sl_2. In the present paper, we prove an improved version of Habiro's conjecture. As an application, we prove a divisibility property of the colored Jones polynomial of boundary links.