Burau representation

In mathematics the Burau representation is a representation of the braid groups, named after and originally studied by the German mathematician Werner Burau during the 1930s. The Burau representation has two common and near-equivalent formulations, the reduced and unreduced Burau representations.
Definition.
Consider the braid group to be the mapping class group of a disc with marked points . The homology group is free abelian of rank . Moreover, the invariant subspace of (under the action of ) is primitive and infinite cyclic. Let be the projection onto this invariant subspace. Then there is a covering space corresponding to this projection map. Much like in the construction of the Alexander polynomial, consider as a module over the group-ring of covering transformations , which is isomorphic to the ring of Laurent polynomials . As a -module, is free of rank . By the basic theory of covering spaces, acts on , and this representation is called the "reduced Burau representation".
The "unreduced Burau representation" has a similar definition, namely one replaces with its (real, oriented) blow-up at the marked points. Then instead of considering one considers the relative homology where is the part of the boundary of corresponding to the blow-up operation together with one point on the disc's boundary. denotes the lift of to . As a -module this is free of rank .
By the homology long exact sequence of a pair, the Burau representations fit into a short exact sequence
where (resp. ) is the reduced (resp. unreduced) Burau -module and is the complement to the diagonal subspace, in other words:
and acts on by the permutation representation.
Explicit matrices.
Let denote the standard generators of the braid group . Then the unreduced Burau representation may be given explicitly by mapping
for , where denotes the identity matrix. Likewise, for the reduced Burau representation is given by
while for , it maps
Bowling alley interpretation.
Vaughan Jones gave the following interpretation of the unreduced Burau representation of positive braids for in – i.e. for braids that are words in the standard braid group generators containing no inverses – which follows immediately from the above explicit description:
Given a positive braid on strands, interpret it as a bowling alley with intertwining lanes. Now throw a bowling ball down one of the lanes and assume that at every crossing where its path crosses over another lane, it falls down with probability and continues along the lower lane. Then the 'th entry of the unreduced Burau representation of is the probability that a ball thrown into the 'th lane ends up in the 'th lane.
Relation to the Alexander polynomial.
If a knot is the closure of a braid in , then, up to multiplication by a unit in , the Alexander polynomial of is given by
where is the reduced Burau representation of the braid .
For example, if in , one finds by using the explicit matrices above that
and the closure of is the unknot whose Alexander polynomial is .
Faithfulness.
The first nonfaithful Burau representations were found by John A. Moody without the use of computer, using a notion of winding number or contour integration. A more conceptual understanding, due to Darren D. Long and Mark Paton interprets the linking or winding as coming from Poincaré duality in first homology relative to the basepoint of a covering space, and uses the intersection form (traditionally called Squier's Form as Craig Squier was the first to explore its properties). Stephen Bigelow combined computer techniques and the Long–Paton theorem to show that the Burau representation is not faithful for . Bigelow moreover provides an explicit non-trivial element in the kernel as a word in the standard generators of the braid group: let
Then an element of the kernel is given by the commutator
The Burau representation for has been known to be faithful for some time. The faithfulness of the Burau representation when is an open problem. The Burau representation appears as a summand of the Jones representation, and for , the faithfulness of the Burau representation is equivalent to that of the Jones representation, which on the other hand is related to the question of whether or not the Jones polynomial is an unknot detector.
Geometry.
Craig Squier showed that the Burau representation preserves a sesquilinear form. Moreover, when the variable is chosen to be a transcendental unit complex number near , it is a positive-definite Hermitian pairing. Thus the Burau representation of the braid group can be thought of as a map into the unitary group U("n").