An application of braid group theory to the finite time dead-core rate

We consider the dead-core problem for the semilinear heat equation with strong absorption and with positive boundary values in a ball. We investigate the dead-core rate, i.e. the rate at which the solution reaches its first zero. We first show, as in the one-dimensional case, that the dead-core rate is always faster than the self-similar rate. By using some special solutions and the braid group theory, we then derive the exact dead-core rates for a large class of initial data.     

On Krammer's representation of the braid group

A connection is made between the Krammer representation and the Birman-Murakami-Wenzl algebra. Inspired by a dimension argument, a basis is found for a certain irrep of the algebra, and relations which generate the matrices are found. Following a rescaling and change of parameters, the matrices are found to be identical to those of the Krammer representation. The two representations are thus the same, proving the irreducibility of one and the faithfulness of the other. Received: 16 April 2000 / Published online: 23 July 2001     

Two-generator subgroups of the pure braid group

We show that any two elements of the pure braid group either commute or generate a free group, settling a question of Luis Paris. Our proof involves the theory of 3-manifolds and the theory of group actions on trees.     

The braid group of a necklace

We show several geometric and algebraic aspects of a necklace: a link composed with a core circle and a series of (unlinked) circles linked to this core. We first prove that the fundamental group of the configuration space of necklaces (that we will call braid group of a necklace) is isomorphic to the braid group over an annulus quotiented by the square of the center. We then define braid groups of necklaces and affine braid groups of type \(\mathcal {A}\) in terms of automorphisms of free groups and characterize these automorphisms among all automorphisms of free groups. In the case of affine braid groups of type \(\mathcal {A}\) such a representation is faithful.     

Generalised Magnus modules over the braid group

Supported byDeutsche Forschungsgemeinschaft     

An extension of Nori fundamental group

In this paper, we study a certain extension of Nori’s fundamental group in the case where a base field is of characteristic 0 and give structure theorems about it. As a result for a smooth projective curve with genus g>1, we prove that Nori’s fundamental group acts faithfully on the category of unipotent bundles on the universal covering. In the case when g = 1, we give a more finer result.     

Finite quotients of the pure symplectic braid group

We show how the finite symplectic groups arise as quotients of the pure symplectic braid group. Via [SV] certain of these groups — in particular, all groups Sp _n (2) — occur as Galois groups over ℚ. Supported by NSF grant DMS-9306479.     

The cohomology ring of the colored braid group

The cohomology ring is obtained for the space of ordered sets of n different points of a plane.Translated from Matematicheskie Zametki, Vol. 5, No. 2, pp. 227–231, February, 1969.The author thanks V. P. Palamodov and D. B. Fuks for useful discussions.     

The Deligne complex for the four-strand braid group

This paper concerns the homotopy type of hyperplane arrangements associated to infinite Coxeter groups acting as reflection groups on . A long-standing conjecture states that the complement of such an arrangement should be aspherical. Some partial results on this conjecture were previously obtained by the author and M. Davis. In this paper, we extend those results to another class of Coxeter groups. The key technical result is that the spherical Deligne complex for the 4-strand braid group is CAT(1).

     

New subfactors from braid group representations

This paper is about the construction of new examples of pairs of subfactors of the hyperfinite II factor, and the computation of their indices and relative commutants. The construction is done in general by considering unitary braid representations with certain properties that are satisfied in natural examples. We compute the indices explicitly for the particular cases in which the braid representations are obtained in connection with representation theory of Lie algebras of types A,B,C,D.

     

The D n generalized pure braid group

A corepresentation for the generalized pure braid group ID _n of the Coxeter group D _n is constructed. The lower central series of ID _n is investigated. It is proved that ID _n is approximable by torsion-free nilpotent groups, so R. Hain's obstruction to the solvability of the generalized Riemann-Hilbert problem is trivial for ID _n.     

The braid group B4 is linear

We study a linear representation ρ:B _n ? GL _m (Z[q ^±1,t ^±1]) with m=n(n-1)/2. We will show that for n=4, this representation is faithful. We prove a relation with the new Charney length function. We formulate a conjecture implying that ρ is faithful for all n. Oblatum 15-VI-1999 & 24-II-2000?Published online: 18 September 2000     

The pure virtual braid group is quadratic

If an augmented algebra $K$ over $\mathbb Q $ is filtered by powers of its augmentation ideal $I$ , the associated graded algebra $gr_I K$ need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this paper, we give a sufficient criterion (called the PVH Criterion) for $gr_I K$ to be quadratic. When $K$ is the group algebra of a group $G$ , quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for $G$ . Thus, the PVH Criterion also implies the existence of such a universal finite type invariant for the group $G$ . We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a universal finite type invariant.     

A generalized Desargues configuration and the pure braid group

In this paper, a configuration with n = (₂^d) points in the plane is described. This configuration, as a matroid, is a Desargues configuration if d = 5, and the union of (₅^d) such configurations if d> 5. As an oriented matroid, it is a rank 3 truncation of the directed complete graph on d vertices. From this fact, it follows from a version of the Lefschetz-Zariski theorem implied by results of Salvetti that the fundamental group π of the complexification of its line arrangement is Artin's pure (or coloured) braid group on d strands.

In this paper we obtain, by using techniques introduced by Salvetti, a new algorithm for finding a presentation of π based on this particular configuration.     

Inner products on the Hecke algebra of the braid group

We claim that the Homfly polynomial (that is to say, Ocneanu's trace functional) contains two polynomial-valued inner products on the Hecke algebra representation of Artin's braid group. These bear a close connection to the Morton-Franks-Williams inequality. With respect to these structures, the set of positive, respectively negative permutation braids becomes an orthonormal basis. In the second case, many inner products can be geometrically interpreted through Legendrian fronts and rulings.