A Graphical Construction of the sl(3) Invariant for Virtual Knots

TL;DR

This paper introduces a new graph-based invariant for virtual knots that generalizes existing invariants, providing new tools for analyzing virtual and free knots, and establishing connections with virtual braids and algebraic structures.

Contribution

It generalizes the Kuperberg sl(3) bracket to virtual knots, creating a new invariant that extends classical invariants and enables minimality proofs and new invariant constructions.

Findings

The invariant coincides with the Homflypt sl(3) invariant for classical knots.

It offers new insights into virtual knots and graphs, aiding in minimality theorems.

The Penrose coloring bracket is shown as a special case of the Kuperberg bracket.

Abstract

By generalizing the Kuperberg sl(3) bracket, we construct a graph-valued analogue of the Homflypt sl(3) invariant for virtual knots. The restriction of this invariant for classical knots coincides with the usual Homflypt sl(3) invariant, and for virtual knots and graphs it provides new information that allows one to prove minimality theorems and to construct new invariants for free knots. We formulate this new invariant for virtual braids as well, and show that it leads to the construction of a trace function on the virtual Hecke algebra. Finally, we show that the Penrose coloring bracket is a special case of the Kuperberg bracket, and we raise new questions about the extension of the present work.