$A_2$ colored polynomials of rigid vertex graphs

TL;DR

This paper introduces a new generalization of the Kauffman-Vogel polynomial for 4-valent rigid vertex graphs using the $A_2$ bracket and clasps, linking it to $rak{sl}_3$ colored Jones polynomials for singular knots.

Contribution

It presents an alternative generalization of the polynomial invariants using $A_2$ structures, extending previous work to oriented and unoriented graphs.

Findings

Defines the $A_2$ colored polynomial invariants for rigid vertex graphs.

Connects these invariants to $rak{sl}_3$ colored Jones polynomials.

Provides a new tool for studying invariants of singular knots and links.

Abstract

The Kauffman-Vogel polynomials are three variable polynomial invariants of $4$-valent rigid vertex graphs. A one-variable specialization of the Kauffman-Vogel polynomials for unoriented $4$-valent rigid vertex graphs was given by using the Kauffman bracket and the Jones-Wenzl idempotent colored with $2$. Bataineh, Elhamdadi and Hajij generalized it to any color with even positive integers. We give another generalization of the one-variable Kauffman-Vogel polynomial for oriented and unoriented $4$-valent rigid vertex graphs by using the $A_2$ bracket and the $A_2$ clasps. These polynomial invariants are considered as the $\mathfrak{sl}_3$ colored Jones polynomials for singular knots and links.