Braid theory is the study of braids, their deformations, and their algebraic structure. Two braids on the same number
of strands are considered equivalent when one can be deformed into the other by an
isotopy that keeps the endpoints
fixed and keeps every strand directed downward. Equivalence classes of 
-strand braids form the braid
group 
,
with multiplication given by stacking one braid beneath
another.
Braid theory connects group theory with knot theory. Closing the upper and lower endpoints of a braid produces a knot or link. Alexander's theorem states that every oriented link can be represented as the closure of a braid, while Markov's theorem characterizes when two closed braids represent equivalent links.
