This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety X. The motivation for this is M. Kontsevich’s homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when dimX ≥ 2, our braid group actions are always faithful. We describe conjectural mirror s...

Let M be a compact, connected surface, possibly with a finite set of points removed from its interior. Let d, n P N, and let € M be a d-fold covering space of M. We show that the covering map induces an embedding of the nth braid group BnpMq of M in the dnth braid group Bdnp€ Mq of € M, and give several applications of this result. First, we classify the finite subgroups of the nth braid group ...

The purpose of this article is to describe connections between the loop space of the 2-sphere, and Artin’s braid groups. The current article exploits Lie algebras associated to Vassiliev invariants in work of T. Kohno [19, 20], and provides connections between these various topics. Two consequences are as follows: (1) the homotopy groups of spheres are identified as “natural” sub-quotients of f...

We give presentations of braid groups and pure braid groups on surfaces and we show some properties of surface pure braid groups. 1. Presentations for surface braids Let F be an orientable surface and let P = {P1, . . . , Pn} be a set of n distinct points of F . A geometric braid on F based at P is an n-tuple Ψ = (ψ1, . . . , ψn) of paths ψi : [0, 1] → F such that • ψi(0) = Pi, i = 1 . . . , n;...

We study the problem of determining if the braid group representations obtained from quantum groups of types E,F and G at roots of unity have infinite image or not. In particular we show that when the fusion categories associated with these quantum groups are not weakly integral, the braid group images are infinite. This provides further evidence for a recent conjecture that weak integrality is...

The topological entropy of a braid is the infimum of the entropies of all homeomor-phisms of the disc which have a finite invariant set represented by the braid. When the isotopy class represented by the braid is pseudo-Anosov or is reducible with a pseudo-Anosov component, this entropy is positive. Fried and Kolev proved that the entropy is bounded below by the logarithm of the spectral radius...

We show that the braid group representations associated with the (3, 6)-quotients of the Hecke algebras factor over a finite group. This was known to experts going back to the 1980s, but a proof has never appeared in print. Our proof uses an unpublished quaternionic representation of the braid group due to Goldschmidt and Jones. Possible topological and categorical generalizations are discussed.

We introduce a recoupling theory for virtual braided trees. This recoupling theory can be utilized to incorporate swap gates into anyonic models of quantum computation. 1 Classical Trees Louis Kauffman and Sam Lomonaco reconstructed the Fibonacci model of quantum computation in [7] by applying the 2-strand Temperly-Lieb recoupling and the skein relation to braided trees. (The original Fibonacci...

In this paper we explore natural connections among extraspecial 2-groups, almost-complex structures, unitary representations of the braid group and the Greenberger-Horne-Zeilinger (GHZ) states. We first present new representations of extraspecial 2-groups in terms of almost-complex structures and use them to derive new unitary braid representations as extensions of representations of the extras...