Burau representation

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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[edit]

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

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

where Vr (resp. Vu) is the reduced (resp. unreduced) Burau Bn-module and DZn is the complement to the diagonal subspace, in other words:

D = \left \{ \left (x_1,\cdots,x_n \right ) \in \mathbf{Z}^n : x_1+\cdots+x_n=0 \right \},

and Bn acts on Zn by the permutation representation.

Explicit matrices[edit]

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

\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 ≤ in − 1, where Ik denotes the k × k identity matrix. Likewise, the reduced Burau representation is given by

\sigma_1 \mapsto \left( \begin{array}{cc|c}-t & 1 & 0 \\ 0 & 1 & 0 \\ \hline 0 & 0 & I_{n-3} \end{array} \right),
\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,
\sigma_{n-1} \mapsto \left( \begin{array}{c|cc} I_{n-3} & 0 & 0 \\ \hline 0 & 1 & 0 \\ 0 & t & -t \end{array} \right).

Bowling alley interpretation[edit]

Vaughan Jones[2] gave the following interpretation of the unreduced Burau representation of positive braids for t in [0,1] – i.e. for braids that are words in the standard braid group generators containing no inverses – which follows immediately from the above explicit description:

Given a positive braid σ on n strands, interpret it as a bowling alley with n intertwining lanes. Now throw a bowling ball down one of the lanes and assume that at every crossing where its path crosses over another lane, it falls down with probability t and continues along the lower lane. Then the (i,j)'th entry of the unreduced Burau representation of σ is the probability that a ball thrown into the i'th lane ends up in the j'th lane.

Relation to the Alexander polynomial[edit]

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

\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

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

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

Faithfulness[edit]

The first nonfaithful Burau representations are found without the use of computer, using a notion of winding number or contour integration.[3] A more conceptual understanding [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]

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[edit]

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 can be thought of as a map into the Unitary group.

References[edit]

  1. ^ Burau, Werner (1936). "Über Zopfgruppen und gleichsinnig verdrillte Verkettungen". Abh. Math. Sem. Hamburg 11: 179−186. 
  2. ^ Jones, Vaughan (1987). "Hecke algebra representations of Braid Groups and Link Polynomials". Annals of Mathematics, Second Series. 126, No. 2: 335−388. 
  3. ^ J. Moody, The faithfulness question for the Burau representation, Proc. AMS 1993
  4. ^ D D Long, M Paton, The Burau representation is not faithful for n ≥ 6, Topology 32 (1993)
  5. ^ 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. 
  6. ^ Bigelow, Stephen (1999). "The Burau representation is not faithful for n = 5". Geometry & Topology 3: 397–404. doi:10.2140/gt.1999.3.397. 
  7. ^ S. Bigelow,International Congress of Mathematicians, Beijing, 2002
  8. ^ V. Turaev, Faithful representations of the braid groups, Bourbaki 1999-2000

External links[edit]