Introduction to Braids

Braids may be thought of as number of pieces of string, which we visualize by drawing lines which may cross over or under each other. For example the diagram in Figure 1 shows the simplest braid with 3 strings – the trivial braid (or the identity braid, denoted by 1), with no crossings. Since we use three strings we say that we have a braid in .

Figure 1: the identity braid, 1.

Two adjacent strings can also cross – passing the left string either under or over the right string. In Figure 2, we show a positive crossing between the leftmost two strings (passing the leftmost string under the middle string) and a negative crossing between the rightmost two strings (passing the middle string over the rightmost string).

Figure2: and in .

In general we may have any number of strings. Numbering the strings from left to right, from , we denote a positive crossing of the i-th string with the (i+1)-st string as and a negative crossing between those strings as . Thus the diagrams shown in Figure 2 are denoted by and .

We can create more complicated braids by stacking these elementary braids– joining the bottom of one braid to the top of the next braid. For example, the braid is shown in Figure 3.

Figure3: in .

We want braids with n strings to encapsulate the idea of stings moving freely in space (with the endpoints fixed). Thus we want the two braids and the identity shown in Figure 4 to be thought of as the same (isotopic). The relations hold for any i (that is, any pair of adjacent strings that cross in the same manner as the first two in Figure 4 can be undone). Note that the inverse of a braid can be obtained by reflecting the whole braid in a horizontal line at the bottom of the braid; for instance, reflect the top half of the braid on the left of Figure 4 to give the bottom half.

Exercise: Reflect the braid in Figure 3 in a horizontal line at the bottom of the braid and compare the result with the braid . Show that the concatenation of these two braids (stack the original braid on top of the reflected braid) is the identity braid.

Figure 4:

Two crossings on non-adjacent pairs of strings may be moved past each other. For example, the braids and in are isotopic, as shown in the figure below.

Figure 5:

We say that and commute (that is, their order does not matter:). In general, the elementary braidsand commute if .

Finally, one can see the following isotopy by picking up the 3^rd string (shown in red) and passing it over the crossing between the first two strings.

Figure 6:

In general this can be written as the relation .

All possible isotopies can be obtained by a sequence of these specified isotopies. In general it is not easy to see if two given braids are the same or not, since changing one into the other may require a complicated sequence of isotopies.

In formal language, a presentation of the braid group on n-strings is given by: