The Burau representation is not faithful for n ≥ 6

Long, D. D.; Paton, M.

doi:10.1016/0040-9383(93)90030-y

Cited by 83 publications

(77 citation statements)

References 4 publications

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“…Since the Burau representation of B n is not faithful for n ≥ 5 [Big1,LP,Moo], the proposition implies that the answer to Question B, and hence Question A, is no.…”

Section: Connection With Braid Groupsmentioning

confidence: 98%

Alexander and Thurston norms of fibered 3-manifolds

Dunfield¹

2001

Pacific J. Math.

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For a 3-manifold M , McMullen derived from the Alexander polynomial of M a norm on H 1 (M, R) called the Alexander norm. He showed that the Thurston norm is an upper bound for the Alexander norm. He asked if these two norms were the same when M fibers over the circle. Here, I give examples that show this is not the case. This question relates to the faithfulness of the Gassner representations of the braid groups. The key tool used is the Bieri-Neumann-Strebel invariant, and I show a connection between this invariant and the Alexander polynomial.

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“…Since the Burau representation of B n is not faithful for n ≥ 5 [Big1,LP,Moo], the proposition implies that the answer to Question B, and hence Question A, is no.…”

Section: Connection With Braid Groupsmentioning

confidence: 98%

Alexander and Thurston norms of fibered 3-manifolds

Dunfield¹

2001

Pacific J. Math.

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“…where the equalities to the left hold by (25). After evaluating the sums, simplifying and dividing by µ 12 which is known to be a non-zero scalar, all the terms in (26) simplify nicely.…”

Section: Existence Of a One-dimensional Invariant Subspacementioning

confidence: 99%

“…In the past, several authors had tried to use the (n − 1)-dimensional Burau representation to show the linearity of B n . However, if this representation is faithful for n = 3, it was shown to be unfaithful for n ≥ 5 (see [27], [25], [1]). It is still unknown whether the Burau representation of B 4 is faithful or not.…”

Section: Definitions and Historymentioning

confidence: 99%

REDUCIBILITY OF THE COHEN–WALES REPRESENTATION OF THE ARTIN GROUP OF TYPE D_n

Levaillant

2012

J. Knot Theory Ramifications

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Abstract. Using knot theory, we construct a linear representation of the CGW algebra of type D n . This representation has degree n 2 − n, the number of positive roots of a root system of type D n . We show that the representation is generically irreducible, but that when the parameters of the algebra are related in a certain way, it becomes reducible. As a representation of the Artin group of type D n , this representation is equivalent to the faithful linear representation of Cohen-Wales. We give a reducibility criterion for this representation as well as a conjecture on the semisimplicity of the CGW algebra of type D n . Our proof is computer-assisted using Mathematica.

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“…This was solved by Moody [9], LongPaton [7], and Bigelow [1] in various cases, with the result that Bur n is nontrivial for n 5; the case of n D 4 is still open. Our next main result is that Bur n is in fact quite large for n 6.…”

Section: The Burau Kernelmentioning

confidence: 99%

Infinite generation of the kernels of the Magnus and Burau representations

Church

Farb

2010

Algebr. Geom. Topol.

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Consider the kernel Mag g of the Magnus representation of the Torelli group and the kernel Bur n of the Burau representation of the braid group. We prove that for g ≥ 2 and for n ≥ 6 the groups Mag g and Bur n have infinite rank first homology. As a consequence we conclude that neither group has any finite generating set. The method of proof in each case consists of producing a kind of "Johnson-type" homomorphism to an infinite rank abelian group, and proving the image has infinite rank. For the case of Bur n , we do this with the assistance of a computer calculation.

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The Burau representation is not faithful for n ≥ 6

Cited by 83 publications

References 4 publications

Alexander and Thurston norms of fibered 3-manifolds

Alexander and Thurston norms of fibered 3-manifolds

REDUCIBILITY OF THE COHEN–WALES REPRESENTATION OF THE ARTIN GROUP OF TYPE D_n

Infinite generation of the kernels of the Magnus and Burau representations

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The Burau representation is not faithful for n ≥ 6

Cited by 83 publications

References 4 publications

Alexander and Thurston norms of fibered 3-manifolds

Alexander and Thurston norms of fibered 3-manifolds

REDUCIBILITY OF THE COHEN–WALES REPRESENTATION OF THE ARTIN GROUP OF TYPE Dn

Infinite generation of the kernels of the Magnus and Burau representations

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Product

Resources

About

REDUCIBILITY OF THE COHEN–WALES REPRESENTATION OF THE ARTIN GROUP OF TYPE D_n