The diagrammatic relations appearing in categories naturally allow us to obtain knot invariants by interpreting knot diagrams within the framework of ribbon categories arising from representation theory. Furthermore, these categories yield invariants of 3-manifolds which are important topological tools.

In this talk, we first introduce the category coming from the representation theory of the quantum group associated to $\mathfrak{sl}_4$ with diagrammatic presentations, also known as $SL_4$-spider categories. In particular, we then relate the 3-valent $SL_4$-spider category defined by Cautis-Kamnitzer-Morrison, to a 4-valent spider category defined using the Kauffman-Vogel relations. By building up the map between these two categories, we show the equivalence relation between these two categories.