Narrow Results

We describe an algorithm to compute the Kuperberg invariant of a combed or framed 3–manifold, starting from a presentation of such a manifold in terms of branched standard spines.

We describe an algorithm to compute the Kuperberg invariant of a combed or framed 3–manifold, starting from a presentation of such a manifold in terms of branched standard spines.

We study relations between group cohomologies and 3-manifold invariants. We first give a combinatorial construction of 3-manifold invariants using weight systems. We give examples of weight systems arising from a particular group cohomology. In the second part we show that these invariants can be obtained in a functorial way.

This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this approach is to take pure virtual braids as primary. The generators of the pure virtual braid group are abstract solutions to the algebraic Yang–Baxter equation. This point of view illuminates representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang–Baxter equation. In this categorical framework, the virtual braid group is a natural group associated with the structure of algebraic braiding. We then point out how the category SC is related to categories associated with quantum algebras and Hopf algebras and with quantum invariants of virtual links.

Any finite-dimensional Hopf algebra $H$ is Frobenius and the stable category of $H$-modules is triangulated monoidal. To $H$-comodule algebras we assign triangulated module-categories over the stable category of $H$-modules. These module-categories are generalizations of homotopy and derived categories of modules over a differential graded algebra. We expect that, for suitable $H$, our construction could be a starting point in the program of categorifying quantum invariants of 3-manifolds.

This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics.

We show that any 3D topological quantum field theory satisfying physically reasonable factorizability conditions has associated to it in a natural way a Hopf algebra object in a suitable tensor category. We also show that all objects in the tensor category have the structure of left-left crossed bimodules over the Hopf algebra object. For 4D factorizable topological quantum filed theories, we provide by analogous methods a construction of a Hopf algebra category.