- Burau representation
In
mathematics the Burau representation is a representation of thebraid group s. 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 themapping class group of a disc with "n" marked points P_n. Thehomology 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 acovering space ilde P_n corresponding to this projection map. Much like in the construction of theAlexander 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 unitcomplex number near 1 it is a positive-definite Hermitian pairing, thus the Burau representation can be thought of as a map into theUnitary 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).
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