Conventional three-dimensional crystal lattices are terminated by surfaces, which can demonstrate complex rebonding and rehybridisation, localised strain and dislocation formation. Two dimensional crystal lattices, of which graphene is the archetype, are terminated by lines. The additional available dimension at such interfaces opens up a range of new topological interface possibilities. We show that graphene sheet edges can adopt a range of topological distortions depending on their nature. Rehybridisation, local bond reordering, chemical functionalisation with bulky, charged, or multi-functional groups can lead to edge buckling to relieve strain , folding, rolling  and even tube formation . We discuss the topological possibilities at a 2D graphene edge, and under what circumstances we expect different edge topologies to occur. Density functional calculations are used to explore in more depth different graphene edge types. Finally we examine the effect of this on graphene properties, and lay out some of the challenges graphene designers face if it is to live up to its promise as the defining material of the 21st century.
Authors: Chris Ewels1, V. Ivanovskaya1, Ph. Wagner1, A. Yaya1, A. Zobelli2, M. Heggie3, P. Briddon4, 1 : Institute of Materials, CNRS, University of Nantes, France, 2: LPS, Universite Paris Sud, Orsay, Franc, 3 : Chemistry Department, University of Sussex, Brighton, UK, 4 : University of Newcastle Upon Tyne, UK
 Ph. Wagner, C. Ewels, V. V. Ivanovskaya, P. R. Briddon, A. Pateau, B. Humbert, submitted (2011)
 V. V. Ivanovskaya, Ph. Wagner, A. Zobelli, I. Suarez-Martinez, A. Yaya, C. P. Ewels, submitted (Proc. GraphITA 2011, Wiley, 2011)
 V. V. Ivanovskaya, A. Zobelli, Ph. Wagner, M. Heggie, P. R. Briddon, M. J. Rayson, C. P. Ewels, Phys. Rev. Lett. In press (2011).