Burau representation

The covering space C_n may be thought of concretely as follows: cut the disk along lines from the boundary to the marked points. Take as many copies of the result as there are integers, stack them vertically, and connect them by ramps going from one side of the cut on one level to the other side of the cut on the level below. This procedure is shown here for n = 4; the covering transformations t^±1 act by shifting the space vertically.

Consider the braid group B_n to be the mapping class group of a disc with n marked points D_n. The homology group H₁(D_n) is free abelian of rank n. Moreover, the invariant subspace of H₁(D_n) (under the action of B_n) is primitive and infinite cyclic. Let π : H₁(D_n) → Z be the projection onto this invariant subspace. Then there is a covering space C_n corresponding to this projection map. Much like in the construction of the Alexander polynomial, consider H₁(C_n) as a module over the group-ring of covering transformations Z[Z], which is isomorphic to the ring of Laurent polynomials Z[t, t⁻¹]. As a Z[t, t⁻¹]-module, H₁(C_n) is free of rank n − 1. By the basic theory of covering spaces, B_n acts on H₁(C_n), and this representation is called the reduced Burau representation.

The unreduced Burau representation has a similar definition, namely one replaces D_n with its (real, oriented) blow-up at the marked points. Then instead of considering H₁(C_n) one considers the relative homology H₁(C_n, Γ) where γ ⊂ D_n is the part of the boundary of D_n corresponding to the blow-up operation together with one point on the disc's boundary. Γ denotes the lift of γ to C_n. As a Z[t, t⁻¹]-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 → V_r → V_u → D ⊕ Z[t, t⁻¹] → 0,

where V_r (resp. V_u) is the reduced (resp. unreduced) Burau B_n-module and D ⊂ Zⁿ is the complement to the diagonal subspace, in other words:

${\displaystyle D=\left\{\left(x_{1},\cdots ,x_{n}\right)\in \mathbf {Z} ^{n}:x_{1}+\cdots +x_{n}=0\right\},}$

and B_n acts on Zⁿ by the permutation representation.

Let σ_i denote the standard generators of the braid group B_n. Then the unreduced Burau representation may be given explicitly by mapping

${\displaystyle \sigma _{i}\mapsto \left({\begin{array}{c|cc|c}I_{i-1}&0&0&0\\\hline 0&1-t&t&0\\0&1&0&0\\\hline 0&0&0&I_{n-i-1}\end{array}}\right),}$

for 1 ≤ i ≤ n − 1, where I_k denotes the k × k identity matrix. Likewise, for n ≥ 3 the reduced Burau representation is given by

${\displaystyle \sigma _{1}\mapsto \left({\begin{array}{cc|c}-t&1&0\\0&1&0\\\hline 0&0&I_{n-3}\end{array}}\right),}$

${\displaystyle \sigma _{i}\mapsto \left({\begin{array}{c|ccc|c}I_{i-2}&0&0&0&0\\\hline 0&1&0&0&0\\0&t&-t&1&0\\0&0&0&1&0\\\hline 0&0&0&0&I_{n-i-2}\end{array}}\right),\quad 2\leq i\leq n-2,}$

${\displaystyle \sigma _{n-1}\mapsto \left({\begin{array}{c|cc}I_{n-3}&0&0\\\hline 0&1&0\\0&t&-t\end{array}}\right),}$

while for n = 2, it maps

${\displaystyle \sigma _{1}\mapsto \left(-t\right).}$

If a knot K is the closure of a braid f in B_n, then, up to multiplication by a unit in Z[t, t⁻¹], the Alexander polynomial Δ_K(t) of K is given by

${\displaystyle {\frac {1-t}{1-t^{n}}}\det(I-f_{*}),}$

where f_∗ is the reduced Burau representation of the braid f.

For example, if f = σ₁σ₂ in B₃, one finds by using the explicit matrices above that

${\displaystyle {\frac {1-t}{1-t^{n}}}\det(I-f_{*})=t,}$

and the closure of f_* is the unknot whose Alexander polynomial is 1.

The first nonfaithful Burau representations were found by John A. Moody without the use of computer, using a notion of winding number or contour integration.[3] A more conceptual understanding, due to Darren D. Long and Mark Paton[4] 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).[5] Stephen Bigelow combined computer techniques and the Long–Paton theorem to show that the Burau representation is not faithful for n ≥ 5.[6][7][8] Bigelow moreover provides an explicit non-trivial element in the kernel as a word in the standard generators of the braid group: let

${\displaystyle \psi _{1}=\sigma _{3}^{-1}\sigma _{2}\sigma _{1}^{2}\sigma _{2}\sigma _{4}^{3}\sigma _{3}\sigma _{2},\quad \psi _{2}=\sigma _{4}^{-1}\sigma _{3}\sigma _{2}\sigma _{1}^{-2}\sigma _{2}\sigma _{1}^{2}\sigma _{2}^{2}\sigma _{1}\sigma _{4}^{5}.}$

Then an element of the kernel is given by the commutator

${\displaystyle [\psi _{1}^{-1}\sigma _{4}\psi _{1},\psi _{2}^{-1}\sigma _{4}\sigma _{3}\sigma _{2}\sigma _{1}^{2}\sigma _{2}\sigma _{3}\sigma _{4}\psi _{2}].}$

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. The Burau representation appears as a summand of the Jones representation, and for n = 4, the faithfulness of the Burau representation is equivalent to that of the Jones representation, which on the other hand is related to the question of whether or not the Jones polynomial is an unknot detector.[9]

Craig Squier showed that the Burau representation preserves a sesquilinear form.[5] 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 of the braid group B_n can be thought of as a map into the unitary group U(n).

• "Burau's Theorem", The Knot Atlas.