Title:

Braid variants and their applications

In Part I we develop the theory of arcbraids and arclinks, which are generalisations of the usual notions of braids and links; an alternative name for arclinks is irrational tangles. A cubical set without degeneracies is called a Dset. Just as braids induce rack automorphisms, arcbraids induce rack homomorphisms. We show that the formulae of the homomorphism induced by certain arcbraids is identical to that of the face maps of □sets. Thus we can model the face maps of a Oset by arcbraids. However, there are many other arcbraids that do not model the usual face maps. We give a method for constructing new Qsets, with unusual face maps, from arcbraids. Using this method, we construct three Qsets. An alternating sum of the face maps of a □set is the boundary operator of the chain complex associated to the classifying space of the Dset. So, in theory, new formulae for face maps could give rise to new homology theories. We show that quasi Omaps, a generalisation of □maps, induce homeomorphisms of the corresponding classifying spaces. Furthermore, we show that we can form quasi Qmaps between the three Osets constructed. Unfortunately, this confounds the hope for new homology theories, but only in this case! In Part IIwe define the Welded Jones polynomial, which is a nontrivial, welded isotopy invariant of welded links. In Chapter 5, signed Gauss codes are related to the fundamental rack; we give algorithms to compute the effects of operations such as reversing, mirroring, crossing changing and smoothing on these objects. We recall that a signed Gauss code corresponds to a virtual link. In Chapter 6 we show that permuting consecutive o’s in the code is equivalent to the extra isotopy move required for welded links. This allows us to define the Welded Bracket polynomial, which is actually a quotient of the Bracket polynomial of virtual links, and the Welded Jones polynomial can be obtained from this. We give nontrivial examples of computations which distinguish welded links. A theorem of Jones for classical knots, which does not hold for virtual or welded knots, implies that the Welded Jones polynomial is trivial for classical knots. A slight modification leads to the Welded Wpolynomial, which is a nontrivial, welded isotopy invariant of classical knots. We end on the entertaining note that whereas the Jones polynomial of the connected sum of classical knots is the product of the individual polynomials, for the Welded Wpolynomial it is the sum of the individual polynomials.
