EMBEDDING TANGLES IN LINKS

Knots, Links and Tangles

We start with some terminology from differential topology [1]. Let  be a circle and  ≥ 2 be an integer. An immersion  :  → R is a smooth function whose derivative never vanishes. An embedding  :  → R is an immersion that is oneto-one. It follows that () is a manifold but () need not be ( is only locally one-to-one, so consider the map that twists  into a figure eight). A knot is a s...

Knots , Links and Tangles Steven

Tangles for Knots and Links

It is often useful to discuss only small “pieces” of a link or a link diagram while disregarding everything else. For example, the Reidemeister moves describe manipulations surrounding at most 3 crossings, and the skein relations for the Jones and Conway polynomials discuss modifications on one crossing at a time. Tangles may be thought of as small pieces or a local pictures of knots or links, ...

The Configuration Space Integral for Links and Tangles in R

The perturbative expression of Chern-Simons theory for links in Euclidean 3-space is a linear combination of integrals on configuration spaces. This has successively been studied by Guadagnini, Martellini and Mintchev, Bar-Natan, Kontsevich, Bott and Taubes, D. Thurston, Altschuler and Freidel, Yang. . .We give a self-contained version of this study with a new choice of compactification, and we...

Open-closed TQFTs extend Khovanov homology from links to tangles

We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov homology from links to arbitrary tangles, not necessarily even. For every plane diagram of an oriented tangle, we construct a chain complex whose homology is invariant under Reidemeister moves. The terms of this chain complex are modules of a suita...