An Extension of the $sl(n)$ Polynomial to Knotted 4-Valent Graphs

TL;DR

This paper extends the $sl(n)$ polynomial invariant from classical knots to knotted 4-valent graphs using a graphical calculus, providing a new invariant and a minimal set of moves for diagram equivalence.

Contribution

It introduces an extension of the $sl(n)$ polynomial to knotted 4-valent graphs and identifies a minimal generating set of Reidemeister-type moves.

Findings

Constructed an invariant for knotted 4-valent graphs.

Extended the $sl(n)$ polynomial to new graph types.

Provided a minimal set of moves for diagram equivalence.

Abstract

We use planar 4-valent graphs and a graphical calculus involving such graphs to construct an invariant for balanced-oriented, knotted 4-valent graphs. Our invariant is an extension of the $sl(n)$ polynomial for classical knots and links. We also provide a minimal generating set of Reidemeister-type moves for diagrams of balanced-oriented, knotted 4-valent graphs.