Knot Diagrams and Reidemeister Moves

It is assumed throughout that a (classical) knot is a smooth embedding $f: \mathbb{S}^1 \rightarrow \mathbb{S}^3$ or $\mathbb{R}^3$, where $\mathbb{S}^n$ denotes the $n$-dimensional sphere, $\mathbb{S}^n=\{ x \in \mathbb{R}^{n+1} \vert \vert x \vert = 1 \}$. If no confusion arises, we also mean the image $K=f(\mathbb{S}^1)$ by a knot. The equivalence of knots is by smooth ambient isotopies. Another set-up commonly used is in the Piecewise-Linear (PL), locally flat category. In general knot theory is study of embeddings.

Instead of going through detailed set-up, such as detailed definitions involved and basic and fundamental theorems used for smooth manifolds, their embeddings and projections, we take a common practice where we take the results on Reidemeister moves, and take combinatorially represented knot diagrams as our subject of study, and Reidemeister moves as their equivalence. Furthermore, such diagrammatic approaches are explained in details in many books in knot theory, so the purpose of this chapter is to briefly review this combinatorial set-up and establish our conventions.