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 with the color 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 A₂ bracket and the A₂ clasps. These polynomial invariants are considered as the $\mathfrak{sl}_3$ colored Jones polynomials for singular knots and links.

The author would like to express his gratitude to his adviser, Hisaaki Endo, for his encouragement.