User:Rybu/Burau

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 $$\pi : H_1 P_n \to \mathbb Z$$ 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 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 $$H_1 \tilde P_n$$ 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 (ie: $$D = \{(x_1,\cdots,x_n) \in \mathbb Z^n : x_1+x_2+\cdots+x_n=0\}$$, 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
Stephen Bigelow showed that the Burau representation not faithful provided $$ n \geq 5$$. 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-definate Hermitian pairing, thus the Burau representation can be thought of as a map into the Unitary group.