On the universal sl_2 invariant of Brunnian bottom tangles

TL;DR

This paper studies the universal sl_2 invariant for Brunnian bottom tangles, identifying a specific subalgebra where it resides and exploring implications for the colored Jones polynomial of Brunnian links.

Contribution

It introduces a small subalgebra of the completed tensor power of U_h(sl_2) where the invariant of Brunnian bottom tangles is contained, and applies this to colored Jones polynomial divisibility.

Findings

Universal sl_2 invariant of Brunnian bottom tangles lies in a specific subalgebra.

Provides a divisibility property for the colored Jones polynomial of Brunnian links.

Establishes a new algebraic framework for analyzing Brunnian tangles.

Abstract

A link L is called Brunnian if every proper sublink of L is trivial. Similarly, a bottom tangle T is called Brunnian if every proper subtangle of T is trivial. In this paper, we give a small subalgebra of the n-fold completed tensor power of U_h(sl_2) in which the universal sl_2 invariant of n-component Brunnian bottom tangles takes values. As an application, we give a divisibility property of the colored Jones polynomial of Brunnian links.