COMBIKNOT · Combinatorial knot theory
FP6 — Marie Curie Actions (Human Resources and Mobility)
- Duration
- 2006-09-01 → 2008-05-31
- EU contribution
- €141,674
- Participants
- 1
- Scheme
- IIF
Lines connect the coordinator with its partners.
Results in brief
Final Activity Report Summary - COMBIKNOT (Combinatorial knot theory)
The project was concerned with knot theory, studying topological features of closed curves in three-dimensional space that were treated as rather stretchable pieces of physical rope. The focus was primarily combinatorial, with data about a curve being encoded in some finite symbolic sequence. The key problem was to achieve this in a way enabling to recognise when two encodings arose from different appearances of the same curve. % One major piece of work in this project was the development of techniques to handle the information originating from a controlled series of curve views, known as one-parameter knot theory, rather than from a standard single view. This was reported in the joint papers with Fiedler, and was used for further work by the time of the project completion. The project also developed algebraic techniques to simplify the analysis of combinatorial data arising in related areas. These included work on three-page embeddings of graphs, Gauss diagrams for links and link groups. Further algebraic results were achieved in establishing a compressed version of the Baker-Campbell-Hausdorff formula for Lie algebras.
Data: CORDIS, © European Union
Project objective
The aim is to study curves and surfaces in 3-space, using discrete combinatorial data including braid and polygonal descriptions of curves and graphs. Polygonal descriptions will be used to capture features of curves and surfaces, which have a small number of extreme points measured in all spatial directions. Building on his previous successful visit to Liverpool under the INTAS scheme, recent methods of 3-page presentations for curves and graphs, familiar to the fellow, will be explored further. The interplay of these methods with braid techniques will allow the fellow to diversify and extend his knowledge of braids and knot invariants. Investigations will be directed in the first instance at embedded trivalent graphs having very few local maxima, and at knots, which can be realised as closed braids on 4 strings, and properties of their Jones polynomials.
Original text from CORDIS.
Participants
- THE UNIVERSITY OF LIVERPOOL · LIVERPOOLCoordinatorUnited Kingdom
Links
Data: CORDIS, © European Union