Jacobi identities in low-dimensional topology

Jacobi identities in low-dimensional topology

The Jacobi identity is the key relation in the definition of a Lie algebra. In the last decade, it also appeared at the heart of the theory of finite type invariants of knots, links and 3-manifolds (and is there called the IHX-relation). In addition, this relation was recently found to arise naturally in a theory of embedding obstructions for 2-spheres in 4-manifolds [20]. We expose the underly...

Problems in Low-dimensional Topology

Four-dimensional topology is in an unsettled state: a great deal is known, but the largest-scale patterns and basic unifying themes are not yet clear. Kirby has recently completed a massive review of low-dimensional problems [Kirby], and many of the results assembled there are complicated and incomplete. In this paper the focus is on a shorter list of \tool" questions, whose solution could unif...

Low - dimensional Topology 57

Low-dimensional topology and geometry.

A t the core of low-dimensional topology has been the classification of knots and links in the 3-sphere and the classification of 3and 4-dimensional manifolds (see Wikipedia for the definitions of basic topological terms). Beginning with the introduction of hyperbolic geometry into knots and 3-manifolds by W. Thurston in the late 1970s, geometric tools have become vital to the subject. Next cam...

Algorithms for Low-Dimensional Topology

In this article, we re-introduce the so called ”Arkaden-FadenLage” (AFL for short) representation of knots in 3 dimensional space introduced by Kurt Reidemeister and show how it can be used to develop efficient algorithms to compute some important topological knot structures. In particular, we introduce an efficient algorithm to calculate holonomic representation of knots introduced by V. Vassi...