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.