The purpose of a proof is to reduce the complexity of a statement until it becomes a sequence of trivialities. To this end, the choice of notation, diagrams and overall paradigm can aid in conveying large amounts of information in a simple manner. This compilation thesis focuses on the choice of visual tools to convey algebraic results in the context of mathematical physics, using a categorical paradigm with various topological semantics. The topics range from covering known results in knot theory, abstract diagram categories and low-dimensional topological quantum field theory, to novel results such as the topological rack exclusion principle, tetrahedral symmetry of framed associators and new diagrammatics for graded-monoidal categories based on the Kleisli presentation.We demonstrate how these diagrammatic methods can be used to simplify algebraic proofs and communicate across disciplines.