🛠️ Usage Guide

Installation

If you have make installed, you can launch the demo using:

make demo

This should launch an interactive demo in the console. It is not fully interactive, please read the instruction at each steps

To create a braid, you need to define operations called Artin generators, representing crossings of two strands.

Imagine a 3-strand braid:

Move strand 1 over strand 2:

first_upcrossing = Braid([+1], n_strands=3)

Move strand 1 under strand 2:

first_downcrossing = Braid([-1], n_strands=3)

This is the inverse of the first:

first_downcrossing = first_upcrossing.inverse()

The inverse can be seen as the image of the braid in a mirror. We will see that this holds for any braid.

first_crossing.draw()

This shows strand 1 going over strand 2. Intermediate steps are shown with an arrow from the braid moving abobe the other one:

1 2 3
1>2 3
2 1 3

Move the new middle strand (strand 2) below strand 3:

second_crossing = Braid([-2], n_strands=3)

Using a list:

b = Braid([1, -2], n_strands=3)

Or by multiplying:

basic_step = first_upcrossing * second_crossing

The resulting braid is:

As you may, with this typical braid, if we reepeat the basic_step three times we return to original order of strands:

braid = basic_step * basic_step * basic_step

Or simply:

braid = basic_step ** 3
braid.draw()

Visual:

This is called a pure braid. A pure braid restores the original strand order.

You can check this with:

braid.is_pure()

Now consider applying handle reduction on a series of braids:

for n in range(2):
    b1 = Braid([+n+1], n_strands=3)
    b2 = Braid([-n], n_strands=3)