Braid Theory

Prerequisite: algebra 2

Got more to share from Alexei Sossinsky's Knots: Mathematics with a Twist~

J. W. H. Alexander figured that all knots can be represented as a closed braid. On the left you have a braid between the dashed lines, closed in such a way to make a knot. On the right you have the reverse, a knot split open to make a braid.

So the idea came about that maybe braids can help classify knots, except that not all knots are conveniently coiled. Before we go any further, some terminology:

smoothing: operation on an intersection

orientation (a): curve direction (indicated by arrows)
country (b): curve bounded regions
infinite region (c): the space outside the knot, also considered a country
Seifert circle: smoothed country
desingularization: to make planar representation of a knot with smoothings

nested: two Seifert circles of same orientation inside one another
change of infinity: to bring a point in an unnested Seifert circle out to the infinite region (top right quadrant)

in turmoil: country with two edges of different Seifert circles with same orientation about the country (indicated with weighted arrows)
perestroika: to make new crossings on country in turmoil (bottom right quadrant)

With all that, Pierre Vogel made the Vogel Algorithm, which can coil any knot for making braids. It helps to digitize knot computations as follows:

do smoothing

if in turmoil

--> do perestroika

--> do smoothing

if unnested

--> do change of infinity

done, you have a coiled knot~

(cut it into a braid if you like)

So can braids classify knots? Maybe. Kind of. Not sure.

Let us examine some product properties of braids:

The way of multiplying braids is shown on the left, where you join the bottom of a braid to the top of another. If you think about it thoroughly enough, you can agree that this operation is associative but not commutative. In the middle you have a braid B joined to its mirror image B^-1, which yields the trivial braid e. Note that any braid B joined to the trivial braid is still braid B just as in multiplying by 1.

There is a system of algebraic coding for naming braids, based on elementary braids. The negative superscript indicates an undercrossing.

The subscripts tell which column the crossing is at. The algebraic code of the braid in this example is written at the base.

But there are cases such as these where it does not matter which elementary braid you list first. In this example, i = 1 and j = 3, in which | 1 - 3 | = 2. It is commutative.

Emil Artin went further with this three-braid isotopy. Jeez, what do we do about it. Braids are almost as finicky as knots..