Английская Википедия:Burau representation

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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

Файл:InfiniteCyclicCovering.gif

The covering space Шаблон: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 Шаблон:Math; the covering transformations Шаблон:Math act by shifting the space vertically.

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

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

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

Шаблон:Math

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

<math>D = \left \{ \left (x_1,\cdots,x_n \right ) \in \mathbf{Z}^n : x_1+\cdots+x_n=0 \right \},</math>

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

Explicit matrices

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

<math>\sigma_i \mapsto \left( \begin{array}{c|cc|c} I_{i-1} & 0 & 0 & 0 \\ \hline 0 & 1-t & t & 0 \\ 0 & 1 & 0 & 0 \\ \hline 0 & 0 & 0 & I_{n-i-1} \end{array} \right),</math>

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

<math>\sigma_1 \mapsto \left( \begin{array}{cc|c}-t & 1 & 0 \\ 0 & 1 & 0 \\ \hline 0 & 0 & I_{n-3} \end{array} \right),</math>

<math>\sigma_i \mapsto \left( \begin{array}{c|ccc|c} I_{i-2} & 0 & 0 & 0 & 0 \\ \hline 0 & 1 & 0 & 0 & 0 \\ 0 & t & -t & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \hline 0 & 0 & 0 & 0 & I_{n-i-2} \end{array} \right), \quad 2 \leq i \leq n-2,</math>

<math>\sigma_{n-1} \mapsto \left( \begin{array}{c|cc} I_{n-3} & 0 & 0 \\ \hline 0 & 1 & 0 \\ 0 & t & -t \end{array} \right),</math>

while for Шаблон:Math, it maps

<math>\sigma_1 \mapsto \left( -t \right).</math>

Bowling alley interpretation

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

Relation to the Alexander polynomial

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

<math>\frac{1-t}{1-t^n} \det(I-f_*),</math>

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

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

<math>\frac{1-t}{1-t^n} \det(I-f_*) = t,</math>

and the closure of Шаблон:Math is the unknot whose Alexander polynomial is Шаблон: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 Шаблон:Math. Bigelow moreover provides an explicit non-trivial element in the kernel as a word in the standard generators of the braid group: let

<math>\psi_1 = \sigma_3^{-1}\sigma_2 \sigma_1^2 \sigma_2 \sigma_4^3 \sigma_3 \sigma_2, \quad \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.</math>

Then an element of the kernel is given by the commutator

<math>[\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].</math>

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

External links

• Шаблон:Knot Atlas