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

The covering space Template:Math may be thought of concretely as follows: cut the disk along lines from the boundary to the marked points. Take as many copies of the result as there are integers, stack them vertically, and connect them by ramps going from one side of the cut on one level to the other side of the cut on the level below. This procedure is shown here for Template:Math; the covering transformations Template:Math act by shifting the space vertically.

Consider the braid group Template:Math to be the mapping class group of a disc with Template:Mvar marked points Template:Math. The homology group Template:Math is free abelian of rank Template:Mvar. Moreover, the invariant subspace of Template:Math (under the action of Template:Math) is primitive and infinite cyclic. Let Template:Math be the projection onto this invariant subspace. Then there is a covering space Template:Math corresponding to this projection map. Much like in the construction of the Alexander polynomial, consider Template:Math as a module over the group-ring of covering transformations Template:Math, which is isomorphic to the ring of Laurent polynomials Template:Math. As a Template:Math-module, Template:Math is free of rank Template:Math. By the basic theory of covering spaces, Template:Math acts on Template:Math, and this representation is called the reduced Burau representation.

The unreduced Burau representation has a similar definition, namely one replaces Template:Math with its (real, oriented) blow-up at the marked points. Then instead of considering Template:Math one considers the relative homology Template:Math where Template:Math is the part of the boundary of Template:Math corresponding to the blow-up operation together with one point on the disc's boundary. Template:Math denotes the lift of Template:Math to Template:Math. As a Template:Math-module this is free of rank Template:Mvar.

By the homology long exact sequence of a pair, the Burau representations fit into a short exact sequence

Template:Math

where Template:Math (resp. Template:Math) is the reduced (resp. unreduced) Burau Template:Math-module and Template:Math is the complement to the diagonal subspace, in other words:

$D=\{(x_1,\cdots ,x_n)\in {𝐙}^n:x_1+\cdots +x_n=0\},$

and Template:Math acts on Template:Math by the permutation representation.

Explicit matrices

Let Template:Math denote the standard generators of the braid group Template:Math. Then the unreduced Burau representation may be given explicitly by mapping

${\sigma }_i\mapsto \left(\begin{array}{cccc}{I}_{i-1}& 0& 0& 0\\ 0& 1-t& t& 0\\ 0& 1& 0& 0\\ 0& 0& 0& I_{n-i-1}\end{array}\right),$

for Template:Math, where Template:Math denotes the Template:Math identity matrix. Likewise, for Template:Math the reduced Burau representation is given by

${\sigma }_1\mapsto \left(\begin{array}{ccc}-t& 1& 0\\ 0& 1& 0\\ 0& 0& I_{n-3}\end{array}\right),$

${\sigma }_i\mapsto \left(\begin{array}{ccccc}{I}_{i-2}& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& t& -t& 1& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& I_{n-i-2}\end{array}\right),2\le i\le n-2,$

${\sigma }_{n-1}\mapsto \left(\begin{array}{ccc}{I}_{n-3}& 0& 0\\ 0& 1& 0\\ 0& t& -t\end{array}\right),$

while for Template:Math, it maps

${\sigma }_1\mapsto (-t).$

Bowling alley interpretation

Vaughan Jones gave the following interpretation of the unreduced Burau representation of positive braids for Template:Math in Template:Math – 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 Template:Math on Template:Math strands, interpret it as a bowling alley with Template:Math 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 Template:Math and continues along the lower lane. Then the Template:Math'th entry of the unreduced Burau representation of Template:Math is the probability that a ball thrown into the Template:Math'th lane ends up in the Template:Math'th lane.

Relation to the Alexander polynomial

If a knot Template:Mvar is the closure of a braid Template:Mvar in Template:Math, then, up to multiplication by a unit in Template:Math, the Alexander polynomial Template:Math of Template:Math is given by

$\frac{1-t}{1-t^n}\det (I-f_{*}),$

where Template:Math is the reduced Burau representation of the braid Template:Mvar.

For example, if Template:Math in Template:Math, one finds by using the explicit matrices above that

$\frac{1-t}{1-t^n}\det (I-f_{*})=1,$

and the closure of Template:Math is the unknot whose Alexander polynomial is Template:Math.

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 Template:Math. Bigelow moreover provides an explicit non-trivial element in the kernel as a word in the standard generators of the braid group: let

${\psi }_1={\sigma }_3^{-1}{\sigma }_2{\sigma }_1^2{\sigma }_2{\sigma }_4^3{\sigma }_3{\sigma }_2,{\psi }_2={\sigma }_4^{-1}{\sigma }_3{\sigma }_2{\sigma }_1^{-2}{\sigma }_2{\sigma }_1^2{\sigma }_2^2{\sigma }_1{\sigma }_4^5.$

Then an element of the kernel is given by the commutator

$[{\psi }_1^{-1}{\sigma }_4{\psi }_1,{\psi }_2^{-1}{\sigma }_4{\sigma }_3{\sigma }_2{\sigma }_1^2{\sigma }_2{\sigma }_3{\sigma }_4{\psi }_2].$

The Burau representation for Template:Math has been known to be faithful for some time. The faithfulness of the Burau representation when Template:Math is an open problem. The Burau representation appears as a summand of the Jones representation, and for Template:Math, 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 Template:Mvar is chosen to be a transcendental unit complex number near Template:Math, it is a positive-definite Hermitian pairing. Thus the Burau representation of the braid group Template:Math can be thought of as a map into the unitary group U(n).

External links

• Template:Knot Atlas