Burau representation

In mathematics the Burau representation is a representation of the braid groups. 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 $pi\; :\; H\_1\; P\_n\; o\; Bbb\; Z$ be the projection onto this invariant subspace. Then there is a covering space $ilde\; P\_n$ corresponding to this projection map. Much like in the construction of the Alexander polynomial, consider $H\_1\; ilde\; P\_n$ as a module over the group-ring of covering transformations $Bbb\; Z\; [Bbb\; Z]\; equiv\; Bbb\; Z\; [t^pm]$ (a Laurent polynomial ring). As such a $Bbb\; Z\; [t^pm]$-module, $H\_1\; ilde\; P\_n$ is free of rank "n" − 1. By the basic theory of covering spaces, $B\_n$ acts on $H\_1\; ilde\; P\_n$, and this representation is called the "reduced Burau representation".

The "reduced 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 $H\_1\; ilde\; P\_n$ one considers the relative homology $H\_1\; (\; ilde\; P\_n,\; ilde\; partial)$ 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. $ilde\; partial$ denotes the lift of $partial$ to $ilde\; P\_n$. As a $Bbb\; Z\; [t^pm]$-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 $0\; o\; V\_r\; o\; V\_u\; o\; D\; oplus\; Bbb\; Z\; [t^pm]\; o\; 0$, where $V\_r$ and $V\_u$ are reduced and unreduced Burau $B\_n$-modules respectively and $D\; subset\; Bbb\; Z^n$ is the complement to the diagonal subspace (ie: $D\; =\; \{(x\_1,cdots,x\_n)\; in\; Bbb\; Z^n\; :\; x\_1+x\_2+cdots+x\_n=0\}$, and $B\_n$ acts on $Bbb\; Z^n$ 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

Stephen Bigelow showed that the Burau representation not faithful provided "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 coming from Blanchfield duality. 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

* Squier, "The Burau representation is unitary." Proc. AMS. 90 (1984).
* Bigelow, "The Burau representation is not faithful for n = 5." Geometry and Topology (1999).