The volume conjecture for augmented knotted trivalent graphs

TL;DR

This paper extends the volume conjecture from knots to augmented knotted trivalent graphs and proves it for this class, also showing it applies to links contained within such graphs.

Contribution

It generalizes the volume conjecture to knotted trivalent graphs and proves it for augmented cases, providing new insights into link and graph relationships.

Findings

Proves the volume conjecture for all augmented knotted trivalent graphs.

Shows any link can be embedded in a link satisfying the volume conjecture.

Establishes a broader applicability of the volume conjecture to complex graph structures.

Abstract

We propose to generalize the volume conjecture to knotted trivalent graphs and we prove the conjecture for all augmented knotted trivalent graphs. As a corollary we find that for any link L there is a link containing L for which the volume conjecture holds.