We begin by laying the groundwork for the Artinian version of the braid group. While braids can be dealt with using a number of different representations and levels of abstraction we will confine ourselves to what can be called the geometric braid groups.

Let \(\E^3\) denote Euclidean \(3\)-space, and let \(\E^2_0\) and \(\E^2_1\) be the parallel planes with \(z\)-coordinates \(0\) and \(1\) respectively. For \(1 \leq i \leq n\), let \(P_i\) and \(Q_i\) be the points with coordinates \((i, 0, 1)\) and \((i, 0, 0)\) respectively such that \(P_1, P_2, \ldots, P_n\) lie on the line \(y=0\) in the upper plane, and \(Q_1, Q_2, \ldots, Q_n\) lie on the line \(y = 0\) in the lower plane.

An \(n\)-braid, specifically a geometric \(n\)-braid, is comprised of \(n\) strands (\(s_1, s_2, \ldots, s_n\)), such that \(s_i\) connects the point \(P_i\) to the point \(Q_{\pi(i)}\), for some \(\pi\) where \(\pi\) is the permutation of the braid; if \(\pi\) is trivial then the braid is said to be a pure braid. Furthermore:

• Each strand \(s_i\) intersects the plane \(z=t\) exactly once for each \(t \in [0,1]\).

• The strands \(s_1, s_2, \ldots, s_n\) intersect the plane \(z=t\) at \(n\) distinct points for each \(t \in [0,1]\).

Simply, an \(n\)-braid is comprised of \(n\) strands wich cross each other a finite number of times without intersecting, and travle strictly “downwards”.

For \(n\)-braids \(\alpha\) and \(\beta\) there is a natural operation of composition as seen in Figure 1.2.¹Note, in graphical representations of braids we will use “\(*\)” to denote composition of braids while when dealing with braids algebraically we will use the convention of adjacency. The resulting braid \(\alpha \beta\) is constructed by identifying \(Q_i\) of \(\alpha\) with \(P_i\) of \(\beta\), thereby creating continuous strands. This operation defines a group operation on the set of \(n\)-braids.

The group of \(n\)-braids is denoted \(\B_n\) with \(\P\B_n\) denoting the subgroup of \(\B_n\) formed by braids with trivial permutations, \(\pi(i)=i\), called the pure braid group. The identity of \(\B_n\) is the braid consisting of \(n\) parallel strands with no crossings, while the inverse \(\beta^{-1}\) of a braid \(\beta\) is the vertical reflection of \(\beta\).

When considering braids, strands can be deformed continuously without altering the structure of the braid, as can be seen in Figure 1.3 with \(\beta\beta^{-1}=I_n\) where \(I_n\) denotes the identity braid in \(n\)-strands. When dealing with the braids it can be helpful to consider the simplest form of each braid, to this end we can comb the braid meaning we will continuously deform the strands until there are the fewest possible crossings of strands. A braid that is of this simple form can be referred to as a combed braid.

Notice then that any \(n\)-braid can be represented as the composition of a finite number of \(elementary\) braids \(\sigma_1, \ldots, \sigma_{n-1}\) and their inverses where \(\sigma_i\) denotes a braid differentiated from \(I_m\) solely by the \(i\)th strand crossing over the \((i+1)\)th strand. Thus, \(\sigma_i^{-1}\) is the braid where the \(i\)th strand crosses under the \((i+1)\)th strand.

We can then note that given \(i\) and \(j\), if \(i\) and \(j\) differ by more than one, the elementary braids \(\sigma_i\) and \(\sigma_j\) commute. It is not generally the case that arbitrary braids commute in \(\B_n\) for \(n \geq 3\).

Notice here that \(\Delta\) reverses the order of points (\(\pi(i) = 1 + n - i\)), and thus \(\Delta^2\) preserves the order of points (\(\pi(i)=i\)).

Up until now we have been addressing what are called open braids. Howeverm by wrapping a braid once around an axis and identifying \(P_i\) with points \(Q_i\) we get what is called a closed braid. On closed braids we allow the same type of deformations as on open braids, namely those that are continuous without causing the strands to intersect.

The problem of classification of closed braids is a group theoretic one in which two closed braids, \(A\) and \(B\) can be considered equal if an only if \(B = X A X^{-1}\) for some open braid \(X\).