The braid group B4 is linear

By Daan Krammer. 2000

109 citations recorded by the source corpus.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/16287635a7c23c183efdeea6c9bcac1faec142da

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Advances in Cryptology - CRYPTO 2003 (2003, methodology): “The first one F4 [10] reduces the computation to a (sparse) linear algebra problem (from a theoretical point of view the link between solving algebraic systems and linear algebra is very old, e.”
Artin monoids inject in their groups (2001, methodology): “prove in Section 4 that ψis injective. If Γ = An, then GA n is the braid group on n+ 1 strings, and ψ: G + A n → Gl(V) is equivalent to the representation constructed by Bigelow and Krammer in [Big], [Kra1] and [Kra2]. In this case, V has ﬁnite dimension, and the injectivity of ψimplies the injectiv”
Braid groups are linear (2000, methodology): “It therefore came as a pleasant surprise when Krammer [ Kra99 ] proved that another representation of the braid groups is faithful in the case n = 4. The representation Krammer used is essentially the same as one used by Lawrence in [Law90] to give a topological denition of a certain summand of the ”
The Infimum, Supremum, and Geodesic Length of a Braid Conjugacy Class (2000, background): “n ≥ t > s ≥ 1 relations: atsarq = arqats, (t −r)(t− q)(s−r)(s−q) > 0 atsasr = atrats = asratr, n ≥ t > s > r ≥ 1 The terms old and new are due to Krammer, who used the new presentation in [6]. Both the old and new solutions to the word problem are polynomial in word length and braid inde”
The mapping class group of a genus two surface is linear (2000, background): “ndent complex values to q and t, we consider the image as lying in GL( n 2 ,C). Now Ln(∆2) is a scalar matrix λI. This can be seen by looking at the representation as an action on the module of forks [Kra1]. (In fact, λ = q2nt2.) We will now “rescale” the representation Ln so that ∆2 is mapped to th”
Knot and braid invariants from contact homology II (2003, methodology): “We denote the set of arcs modulo isotopy by Pn. For 1 ≤ i,j≤ nwith i6= j, we deﬁne γij ∈ Pn to be the arc from pi to pj which remains in the upper half plane; see Figure 2. The terminology derives in [Kra], where (unoriented) arcs are used to deﬁnethe Lawrence–Krammer representation of Bn. Indeed, a”
GEOMETRIC PRESENTATIONS FOR THE PURE BRAID GROUP (2006, background): “920s [3]. In 1998, Birman, Ko, and Lee [5] gave a more symmetrical presentation for the braid group on a larger generating set that has become fashionable of late (see, for example, [4], [7], [8], or [11]). Our goal is to apply a similar idea to the pure braid group. The standard ﬁnite presentation ”
Braid groups are linear (2004, methodology): “Other definitions include a second homology group [11], [12] and a pictorial approach [3], [11].”
Braids: A Survey (2004, background): “Later, Krammer gave an entirely algebraic definition of one of these representations: λ : Bn → GLr(A) where A = Z[t±1, q±1] and showed that it was faithful for n = 4 [97].”
DOES THE JONES POLYNOMIAL DETECT THE UNKNOT (2000, background): “Krammer [12] has already shown that #4 is linear.”
Automorphisms of Free Groups and Outer Space (2002, background): “Linearity of Aut(F2) is equivalent to the linearity of the 4-strand braid group B4, which was proved by Krammer [74].”
Basic results on braid groups (2010, methodology): “ear It is not possible to write a note on basic properties of braid groupswithout mentioning that braid groups are linear. This was a long-standing open question until it was solved for B4 by Krammer [51], and then for general Bn using topological methods by Bigelow [8]. Shortly after that Krammer g”