Narrow Results

In 1990, John Berge described several families of knots in the three-dimensional sphere which have non-trivial Dehn surgeries yielding lens spaces. We study a subfamily of them from the view point of resolution of singularity of complex curves and surfaces, Kirby calculus in topology of four-dimensional manifolds and A'Campo's divide knot theory.

We use recoupling theory to study the Kauffman bracket skein module of the quaternionic manifold over ℤ[A^±1] localized by inverting all the cyclotomic polynomials. We prove that the skein module is spanned by five elements. Using the quantum invariants of these skein elements and the ℤ₂-homology of the manifold, we determine that they are linearly independent.

In view of the result of Kontsevich, now often called "the fundamental theorem of Vassiliev theory", identifying the graded dual of the associated graded vector space to the space of Vassiliev invariants filtered by degree with the linear span of chord diagrams modulo the "4T-relation" (and in the unframed case, originally considered by Vassiliev, the "1T-" or "isolated chord relation"), it is a problem of some interest to provide a basis for the space of chord diagrams modulo the 4T-relation.

We construct the basis for the vector space spanned by chord diagrams with n chords and m link components, modulo 4T relations for n ≤ 5.

Simple crystallizations are edge-colored graphs representing piecewise linear (PL) 4-manifolds with the property that the 1-skeleton of the associated triangulation equals the 1-skeleton of a 4-simplex. In this paper, we prove that any (simply-connected) PL 4-manifold $M$ admitting a simple crystallization admits a special handlebody decomposition, too; equivalently, $M$ may be represented by a framed link yielding ${𝕊}^3$, with exactly ${\beta }_2(M)$ components (${\beta }_2(M)$ being the second Betti number of $M$). As a consequence, the regular genus of $M$ is proved to be the double of ${\beta }_2(M)$. Moreover, the characterization of any such PL 4-manifold by $k(M)=3{\beta }_2(M)$, where $k(M)$ is the gem-complexity of $M$ (i.e. the non-negative number $p-1$, $2p$ being the minimum order of a crystallization of $M$), implies that both PL invariants gem-complexity and regular genus turn out to be additive within the class of all PL 4-manifolds admitting simple crystallizations (in particular, within the class of all “standard” simply-connected PL 4-manifolds).

For a large class of closed orientable 3-manifolds, we present a polynomial algorithm to go from a triangulation to a framed link presenting the same manifold. The algorithm applies whenever we can get a resoluble gem from the triangulation. Most minimal gems have this property (about 95% of gems in our complete catalogue up to 28 vertices). From the gems which fail, nothing prevents another gem from inducing the same 3-manifold to be resoluble. We leave as an open problem presenting a 3-manifold which is not induced by a resoluble gem. This paper is an application of the theory of gems to 3-manifold topology.

In this paper, we generalize the well-known constructions of invariants of framed links and 3-manifolds based on Hopf algebras to multiplier Hopf algebras. Results include a structural characterization of ribbon multiplier Hopf algebras from which invariants of framed links arise from the usual functorial construction to the ribbon category of modules, and the extension of Hennings construction of invariants of framed links and 3-manifolds obtained from Hopf algebras to algebraic quantum groups equipped with an appropriate trace.

Using methods suggested by the work of Turaev and Viro [11, 12], we provide a detailed construction of topological quantum field theories associated to finite crossed G-sets. Our construction of theories associated to finite groups fills in some details implicit in Dijkgraaf and Witten's [3] discussion of topological gauge theories with finite gauge group, while the theories associated to finite crossed G-sets simultaneously extend Dijkgraaf and Witten's [3] results to 3-manifolds equipped with links and Freyd and Yetter's [5] construction of link invariants from crossed G-sets from links in the 3-sphere to links in arbitrary 3-manifolds. Topological interpretations of the manifold and link invariants associated to these TQFT's are provided. We conclude discussion of our results as a toy model for QFT and of their relation to quantum groups.

In this paper we give a short survey of exotic smooth structures of 4-manifolds. Various methods of constructing exotic structures and their relation to “Shake-Sliceness” and the Zeeman conjecture is discussed.