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 $\pi :H_{1}P_{n}\to {\mathbb {Z}}$ 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 ${\mathbb {Z}}[{\mathbb {Z}}]\equiv {\mathbb {Z}}[t^{\pm }]$ (a Laurent polynomial ring). As such a ${\mathbb {Z}}[t^{\pm }]$-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 ${\mathbb {Z}}[t^{\pm }]$-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 $0\to V_{r}\to V_{u}\to D\oplus {\mathbb {Z}}[t^{\pm }]\to 0$, where $V_{r}$ and $V_{u}$ are reduced and unreduced Burau $B_{n}$-modules respectively and $D\subset {\mathbb {Z}}^{n}$ is the complement to the diagonal subspace (i.e.: $D=\{(x_{1},\cdots ,x_{n})\in {\mathbb {Z}}^{n}:x_{1}+x_{2}+\cdots +x_{n}=0\}$, and $B_{n}$ acts on ${\mathbb {Z}}^{n}$ 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 $\Delta _{K}(t)=\det(I-f_{*})$ 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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