In a link diagram $D$ an arc is a strand segment of the diagram that ends at underpasses of crossings.

An alternating knot or link is one which has a diagram where the crossings of each component alternate between over- and under-crossings.

The complement of a knot (or link) is the 3-dimensional space obtained by removing the knot from ${ℝ}^{3}$. Topologists sometimes prefer to define the knot complement as the 3-dimensional sphere ${S}^{3}$ with a thickened neighbourhood of the knot removed, to ensure that the knot complement is a compact manifold.

A component of a link is one of the disjoint knots of which it is composed.

Two elements $a$, $b$ of a group $G$ are called conjugate if there exists a $c\in G$ such that ${c}^{-1}·a·c=b$.

The fundamental group of a topological space $X$ with basepoint $x$, denoted ${\pi }_{1}\left(X,x\right)$, is the set of all paths in $X$ which start and end at $x$ under the equivalence relation of homotopy. These paths form a group under the following operation: $f·g$ is the path which first travels around $f$ and then around $g$.

A pair consisting of a set of nodes (vertices) and a set of edges. Each edge is an unordered pair of nodes. For our purposes we will not consider graphs that have multiple edges between the same two nodes.

A group presentation by generators and relations consists of a set $A$ of generators and a set $R$ of relations. The relations are of the form ${r}_{1}={r}_{2}$, where ${r}_{1}$ and ${r}_{2}$ are words in the elements of $A$ and their inverses. Formally this means that the group is constructed as the quotient of the free group on $A$ by the normal subgroup generated by all expressions ${r}_{1}·{{r}_{2}}^{-1}$ for ${r}_{1}={r}_{2}$ belonging to $R$.

A homotopy between two continuous maps $f$, $g$ : $X\to Y$ is a continuous map $H$: $X\times \left[0,1\right]\to Y$ such that, for all points $x$ in $X$, we have $H\left(x,0\right)=f\left(x\right)$ and $H\left(x,1\right)=g\left(x\right)$. The second parameter can be thought of as time, so $H$ is a continuous deformation from $f$ (at time $0$) to $g$ (at time $1$).

An isotopy is a homotopy $H$ where, for each fixed $t$, $H\left(x,t\right)$ is a homeomorphism. If $f$ and $g$ : $X\to Y$ are embeddings of $X$ in $Y$, then an ambient isotopy is an isotopy $H$: $Y\times \left[0,1\right]\to Y$ which takes $f$ to $g$. In particular, ambient isotopies will always preserve orientation.

Suppose that $W$ is a submodule of a module $V$ and a linear representation $r$: $G\to \mathrm{GL}\left(V\right)$ is given such that $W$ is invariant under the representation $r$. Then the restriction of $r$ to $W$ is defined to be the map $s$: $G\to \mathrm{GL}\left(W\right)$, where $s\left(g\right)$ is the restriction of $r\left(g\right)$ to $W$. This is well defined as $r\left(g\right)\left(w\right)\in W$ for every $w\in W$ by the invariance of $W$ under $r$.

The valency of a node in a graph is the number of edges which connect to it. If an edge connects to the node at both ends (i.e. it is a loop) then it is counted twice.