ON THE KAUFFMAN–VOGEL AND THE MURAKAMI–OHTSUKI–YAMADA GRAPH POLYNOMIALS

Abstract

This paper consists of three parts. First, we generalize the Jaeger Formula to express the Kauffman–Vogel graph polynomial as a state sum of the Murakami–Ohtsuki–Yamada graph polynomial. Then, we demonstrate that reversing the orientation and the color of a MOY graph along a simple circuit does not change the 𝔰𝔩(N) Murakami–Ohtsuki–Yamada polynomial or the 𝔰𝔩(N) homology of this MOY graph. In fact, reversing the orientation and the color of a component of a colored link only changes the 𝔰𝔩(N) homology by an overall grading shift. Finally, as an application of the first two parts, we prove that the 𝔰𝔬(6) Kauffman polynomial is equal to the 2-colored 𝔰𝔩(4) Reshetikhin–Turaev link polynomial, which implies that the 2-colored 𝔰𝔩(4) link homology categorifies the 𝔰𝔬(6) Kauffman polynomial.