On the universal sl_2 invariant of ribbon bottom tangles

TL;DR

This paper proves that the universal invariant of ribbon bottom tangles, associated with U_h(sl_2), lies within a specific algebraic substructure, with implications for the colored Jones polynomial of ribbon links.

Contribution

It establishes a new algebraic containment property for the universal invariant of ribbon bottom tangles, connecting it to the structure of U_h(sl_2).

Findings

Universal invariant J_T is contained in a Z[q,q^{-1}]-subalgebra.

Result applies to colored Jones polynomial of ribbon links.

Provides algebraic insight into invariants of ribbon bottom tangles.

Abstract

A bottom tangle is a tangle in a cube consisting of arc components whose boundary points are on a line in the bottom square of the cube. A ribbon bottom tangle is a bottom tangle whose closure is a ribbon link. For every n-component ribbon bottom tangle T, we prove that the universal invariant J_T of T associated to the quantized enveloping algebra U_h(sl_2) of the Lie algebra sl_2 is contained in a certain Z[q,q^{-1}]-subalgebra of the n-fold completed tensor power of U_h(sl_2). This result is applied to the colored Jones polynomial of ribbon links.