# 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[1] 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 $\pi : H_1 P_n \to \Bbb Z$ 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 $\Bbb Z[\Bbb Z] \equiv \Bbb Z[t^\pm]$ (a Laurent polynomial ring). As such a $\Bbb Z[t^\pm]$-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 $\Bbb Z[t^\pm]$-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 $0 \to V_r \to V_u \to D \oplus \Bbb Z[t^\pm] \to 0$, 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 (i.e.: $D = \{(x_1,\cdots,x_n) \in \Bbb Z^n : x_1+x_2+\cdots+x_n=0\}$, 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

The first nonfaithful Burau representations are found without the use of computer, using a notion of winding number or contour integration. [2] A more conceptual understanding [3] 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).[4] Stephen Bigelow combined computer techniques and the Long-Paton theorem to show that the Burau representation is not faithful for n ≥ 5.[5] [6][7]

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.[4] 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

1. ^ Burau, Werner (1936). "Über Zopfgruppen und gleichsinnig verdrillte Verkettungen". Abh. Math. Sem. Hamburg 11: 179−186.
2. ^
3. ^ D D Long, M Paton, The Burau representation is not faithful for n ≥ 6, Topology 32 (1993)
4. ^ a b Squier, Craig C (1984). "The Burau representation is unitary". Proceedings of the American Mathematical Society 90 (2): 199–202. doi:10.2307/2045338.
5. ^ Bigelow, Stephen (1999). "The Burau representation is not faithful for n = 5". Geometry & Topology 3: 397–404. doi:10.2140/gt.1999.3.397.
6. ^
7. ^