**UBC Mathematics Department**

*http://www.math.ubc.ca/*

*http://www.math.ubc.ca/~djun/thesis/java/cutseq.html*

This Java applet was written by
Djun M. Kim, based on ideas of Roger Fenn,
Michael Greene, Dale Rolfsen, Colin Rourke, and Bert Wiest. |

## Braids
A
The identity element of the group is the braid in which the strings are straight lines which do not intertwine at all. The inverse of a braid is its reflection in a mirror perpendicular to the preferred direction.
This multiplication gives the set of
i - j |
> 1 when | s _{i} s_{j} s_{i} =
s_{j} s_{i} s_{j}
i - j |
= 1.
PICTURE ## Braids As Dances:
By thinking of the preferred direction as "time", a braid may be
regarded as the time history of j and j+1 exchange places,
rotating clockwise (say), while the other dancers stay put.
PICTURE (ANIMATED?)
Imagining the plane (dance floor) to be made of rubber which sticks to
the dancers as they twirl about. In this way, the motion of the
points in a dance can extend to a continuous motion of the whole dance
floor. The line, which started straight, at the end of the dance
becomes a (possibly) very convoluted curve in the plane, which
nevertheless visits each of the integer points, does not
self-intersect, and is eventually straight (beyond the points 0 and
Just as different-looking braids may really be the same, different curve diagrams may correspond to the same braid. But up to the appropriate notions of equivalence, there is a one-to-one correspondence between braids and curve diagrams. The nice thing about curve diagrams is that there is a unique "canonical form" within each equivalence class. It is the diagram which has the smallest number of components of intersection with the straight line, consists of perfect semicircles in each half-plane above and below the straight line, possibly some straight line segments between adjacent starting points, and so that the curve passes through gaps between the points at equal intervals.
## Ordering Braids
Given two braids, we look at their canonical curve diagrams.
Following the diagrams from left to right, they will coincide for a
while. But if the braids are different, their curve diagrams will
also be different. Looking at the "first" point at which these curves
diverge, one will turn left relative to the other. We will regard the
one which turns left to be the greater of the two. It is not hard to
see that this ordering is "invariant" under right multiplication: if
PICTURE (examples) For more precise definitions, and proofs, please refer to the paper, "Ordering the Braid Groups," by Fenn, Green, Rolfsen, Rourke and Wiest, which may be downloaded from the site: http://xxx.lanl.gov/. |

## This page and its contents (text, programs, images, etc) are copyright ©1998 by Dale Rolfsen and Djun Kim.Return to Djun's Java Projects page |