Topology (electrical circuits)

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 $B_n$ to be the mapping class group of a disc with n marked points $P_n$. The homology group $H_1{P}_n$ is free abelian of rank n. Moreover, the invariant subspace of $H_1{P}_n$ (under the action of $B_n$) is primitive and infinite cyclic. Let $\pi :H_1{P}_n\to \mathbb{\mathbb{Z}}$ be the projection onto this invariant subspace. Then there is a covering space ${\tilde{P}}_n$ corresponding to this projection map. Much like in the construction of the Alexander polynomial, consider $H_1{\tilde{P}}_n$ as a module over the group-ring of covering transformations $\mathbb{\mathbb{Z}}[\mathbb{\mathbb{Z}}]\equiv \mathbb{\mathbb{Z}}[t^{\pm }]$ (a Laurent polynomial ring). As such a $\mathbb{\mathbb{Z}}[t^{\pm }]$-module, $H_1{\tilde{P}}_n$ is free of rank n − 1. By the basic theory of covering spaces, $B_n$ acts on $H_1{\tilde{P}}_n$, and this representation is called the reduced Burau representation.

The unreduced Burau representation has a similar definition, namely one replaces $P_n$ with its (real, oriented) blow-up at the marked points. Then instead of considering $H_1{\tilde{P}}_n$ one considers the relative homology $H_1({\tilde{P}}_n,\tilde{\partial })$ where $\partial \subset P_n$ is the part of the boundary of $P_n$ corresponding to the blow-up operation together with one point on the disc's boundary. $\tilde{\partial }$ denotes the lift of $\partial$ to ${\tilde{P}}_n$. As a $\mathbb{\mathbb{Z}}[t^{\pm }]$-module this is free of rank n.

By the homology long exact sequence of a pair, the Burau representations fit into a short exact sequence $0\to V_r\to V_u\to D\oplus \mathbb{\mathbb{Z}}[t^{\pm }]\to 0$, where $V_r$ and $V_u$ are reduced and unreduced Burau $B_n$-modules respectively and $D\subset {\mathbb{\mathbb{Z}}}^n$ is the complement to the diagonal subspace (i.e.: $D=\{(x_1,\cdots ,x_n)\in {\mathbb{\mathbb{Z}}}^n:x_1+x_2+\cdots +x_n=0\}$, and $B_n$ acts on ${\mathbb{\mathbb{Z}}}^n$ by the permutation representation.

Relation to the Alexander polynomial

If a knot $K$ is the closure of a braid $f$, then the Alexander polynomial is given by ${\mathrm{\Delta }}_K(t)=\det (I-f_{*})$ where $f_{*}$ is the reduced Burau representation of the braid $f$.

Faithfulness

The first nonfaithful Burau representations are found without the use of computer, using a notion of winding number or contour integration. A more conceptual understanding 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 n ≥ 5.

The Burau representation for n = 2, 3 has been known to be faithful for some time. The faithfulness of the Burau representation when n = 4 is an open problem.

Geometry

Squier showed that the Burau representation preserves a sesquilinear form. Moreover, when the variable $t$ is chosen to be a transcendental unit complex number near $1$ it is a positive-definite Hermitian pairing, thus the Burau representation can be thought of as a map into the Unitary group.

References

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