Braid

We study the homotopy types of complements of arrangements of n transverse planes in R 4 , obtaining a complete classification for n ≤ 6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of a 2-arrangement in R 4 is not determined by the cohomology ring, thereby answering a question of Ziegler. The invariants that we use are derived from the characteristic varieties of the complement. The nature of these varieties illustrates the difference between real and complex arrangements.

The last decades have seen a revival of fragmentation in British and American works of fiction that deny linearity, coherence and continuity in favour of disruption, gaps and fissures. Authors such as Ali Smith, David Mitchell and David Shields have sought new ways of representing our global, media-saturated contemporary experience which differ from modernist and postmodernist experimentations from which the writers nevertheless draw inspiration. This volume aims to investigate some of the most important contributions to fragmentary literature from British and American writers since the 1990s, with a particular emphasis on texts released in the twenty-first century. The chapters within examine whether contemporary forms of literary fragmentation constitute a return to the modernist episteme or the fragmented literature of exhaustion of the 1960s, mark a continuity with postmodernist aesthetics or signal a deviation from past models and an attempt to reflect today’s accelerated cultu...

We describe in this paper different combinatorial objects enumerated by q-Bessel functions : skew Ferrers diagrams according to width and area, simple braids according to number of threads and exponent and pairs of permutations according to common descents and inversions. These enumerations allow us to introduce different technics of counting and q-counting. PI 7 QO n n+v

This theoretical paper provides a basis for the optimality of scale-sets by Guigues [6] and the optimal pruning of binary partition trees by Salembier-Garrido [11]. They extract constrained-optimal cuts from a hierarchy of partitions. Firstly, this paper extends their results to a larger family of partitions, namely the braid [9]. Secondly, the paper shows the dependence of valid constraint function values and multiplier values in a Lagrangian optimization framework. Lastly, but most importantly, it also proposes the energetic order and energetic lattice based solutions for the constraint optimization problem. This approach operates on a partition based constraint thus ensuring the existence of a valid multiplier and constraint value.

It is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of the braid index of an alternating link, Yamada showed that the minimum number of Seifert circles over all regular projections of a link equals the braid index. Thus one may conjecture that the number of Seifert circles in a reduced alternating diagram of the link equals the braid index of the link, but this turns out to be false. In this paper we prove the next best thing that one could hope for: we characterise exactly those alternating links for which their braid indices equal the numbers of Seifert circles in their corresponding reduced alternating link diagrams. More specifically, we prove that if D is a reduced alternating link diagram of an alternating link L, then b(L), the braid index of L, equals the number of Seifert circles in D if and on...