Rabgres · Ramification of braid groups and their related structures

Ramification of braid groups and their related structures

In the RABGRES project, the researcher Neha Nanda undertook the fellowship at the Laboratoire de Mathématiques Nicolas Oresme (LMNO), Université de Caen Normandie, under the supervision of the host Professor John Guaschi, focusing on the ramifications of braid groups and their related structures. Braid groups, initially introduced by Emil Artin, have become foundational objects in several branches of mathematics and theoretical physics, including low-dimensional topology, algebraic geometry, dynamical systems, and quantum physics. They are deeply intertwined with knot theory, which studies the classification of knots and has led to the development of important knot invariants such as the Alexander and Jones polynomials. These braid groups, and their generalisations, offer vast avenues of exploration due to their structural richness and their applications in both pure and applied mathematics. One of the project's key directions was the study of Khovanov's twin groups, which serve as a planar analogue of braid groups, focusing on configurations of strands in a plane. These twin groups are essential for understanding doodles—classes of Jordan curves on the sphere that do not have triple intersections—as explored by Fenn and Taylor. A few related groups which are investigated are cactus groups (twin groups embed in the cactus groups), virtual twin groups (related to doodles on closed oriented surfaces), right-angled Coxeter groups (general class of groups to which twin groups belong), right-angled Artin groups (obtained from the presentation of right-angled Coxeter groups), diagram groups (pure twin groups can be viewed as diagram groups). Even though classical braid groups and some of their generalisations have apparently similar presentations and topological interpretations, it is important to be able to distinguish them and understand how they differ algebraically, and it is interesting to explore the above-mentioned groups independently. The main objectives of the project are as follows. 1. Understand better the (pure) twin groups by studying some of their invariants, such as their virtual cohomological dimension, and look for good presentations for them using Fadell-Neuwirth-like short exact sequences. 2. Following the construction of surface braid groups from Artin braid groups, generalise the (pure) twin groups to surface (pure) twin groups, obtain presentations of these groups, and seek doodle-like structures to which they correspond. 3. Construct new (virtual) doodle invariants, some being analogous to those known for knots and links, the aim is to classify (virtual) doodles. The project aimed to uncover deeper algebraic and topological properties of these structures, contributing to a dynamic and rapidly evolving research area.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз