A finite set of generators for the Kauffman bracket skein algebra

If F is a compact orientable surface it is known that the Kauffman bracket skein module of has a multiplicative structure. Our central result is the construction of a finite set of knots which generate the module as an algebra. We can then define an integer valued invariant of compact orientable 3-manifolds which characterizes . Received November 27, 1995; in final form September 29, 1997     

Representing links in 3-manifolds¶ by branched coverings of S3

We introduce a planar coloured-diagram representation of links in 3-manifolds given as branched coverings of the 3-sphere. We also prove an equivalence theorem based on local moves and the existence of a universal configuration for such representation. As an application we give unified proofs of two different results on existence of fibered links in 3-manifolds. Received: 7 April 1997     

On certain classes of hyperbolic 3-manifolds

We study the topological structure and the homeomorphism problem for closed 3-manifolds M(n,k) obtained by pairwise identifications of faces in the boundary of certain polyhedral 3-balls. We prove that they are (n/d)-fold cyclic coverings of the 3-sphere branched over certain hyperbolic links of d+1 components, where d= (n/k). Then we study the closed 3-manifolds obtained by Dehn surgeries on the components of these links. Received: 27 November 1998 / Accepted: 12 May 1999     

Stiefel–Whitney surfaces and the tri-genus of non-orientable 3-manifolds

Every non-orientable 3-manifold M can be expressed as a union of three orientable handlebodies V ₁,V ₂,V ₃ whose interiors are pairwise disjoint. If g _i denotes the genus of ∂V _i and g ₃≤g ₂≤g ₃, then the tri-genus of M is the minimum triple (g ₁,g ₂,g ₃), ordered lexicographically. If the Bockstein of the first Stiefel–Whitney class βw ₁(M)=0, then M has tri-genus (0,2g,g ₃), where g is the minimal genus of a 2-sided Stiefel Whitney surface of M. In this paper it is shown that, if βw ₁(M)&\ne;0, then M has tri-genus (1,2g−1,g ₃), where g is the minimal genus of a (1-sided) Stiefel–Whitney surface. As an application the tri-genus of certain graph manifolds is computed. Received: 28 April 1999     

Représentations du groupe des tresses généralisées dans des groupes de Lie

Let Γ be a Lie group.Then any automorphism of the free group of rank n induces a diffeomorphism of Γ ⁿ . We use this remark and a result of P. Vogel to construct linear representations of a certain automorphism group of the algebraic closure of the free group. This automorphism group is closely related to the string link cobordism group.

Received:     

The LMO-invariant of 3-manifolds of rank one and the Alexander polynomial

We prove that the LMO-invariant of a 3-manifold of rank one is determined by the Alexander polynomial of the manifold, and conversely, that the Alexander polynomial is determined by the LMO-invariant. Furthermore, we show that the Alexander polynomial of a null-homologous knot in a rational homology 3-sphere can be obtained by composing the weight system of the Alexander polynomial with the ?rhus invariant of knots. Received February 14, 2000 / Published online October 11, 2000     

Singhof fillings of closed 3-manifolds

We give a complete classification of all closed, connected 3-manifolds which admit a Singhof filling with any number of solid tori. Received: 15 March 2001 / Revised version: 17 September 2001     

On the super additivity of tunnel number of knots

Abstract. In the present paper, we show a necessary and sufficient condition for knots in (with some side condition) to have the super additivity of tunnel number under connected sum, i.e., . Received: 25 June 1999 / Revised: 24 October 1999 / Published online: 8 May 2000     

On \pi-hyperbolic knots with the same 2-fold branched coverings

We consider the following problem from the Kirby's list (Problem 3.25): Let K be a knot in and M(K) its 2-fold branched covering space. Describe the equivalence class [K] of K in the set of knots under the equivalence relation if is homeomorphic to . It is known that there exist arbitrarily many different hyperbolic knots with the same 2-fold branched coverings, due to mutation along Conway spheres. Thus the most basic class of knots to investigate are knots which do not admit Conway spheres. In this paper we solve the above problem for knots which do not admit Conway spheres, in the following sense: we give upper bounds for the number of knots in the equivalence class [K] of a knot K and we describe how the different knots in the equivalence class of K are related. Received: 3 August 1998 / in final form: 17 June 1999     

On the geography of symplectic 6-manifolds

The article investigates the geography of closed, connected and simply connected, six-dimensional manifolds. It is proved that any triple of integers satisfying some necessary arithmetical restrictions occurs as the Chern triple of such a manifold. The main tools used for producing the examples are the symplectic connected sum and the symplectic blow-up. Received: 28 May 1998 / Revised version: 22 January 1999     

Rokhlin's inequalities for real algebraic curves and¶an extremal property

Rokhlin's inequalities for real algebraic curves and an extremal property for one of the inequalities are derived via a link cobordism approach. Received: 18 February 1997 / Revised version: 23 September 1998     

Regular Legendrian knots and the HOMFLY polynomial of immersed plane curves

We show that every unframed knot type in has a representative obtained by the Legendrian lifting of an immersed plane curve. This gives a positive answer to the question asked by V.I.Arnold in [3]. The Legendrian lifting lowers the framed version of the HOMFLY polynomial [20] to generic plane curves. We prove that the induced polynomial invariant can be completely defined in terms of plane curves only. Moreover it is a genuine, not Laurent, polynomial in the framing variable. This provides an estimate on the Bennequin-Tabachnikov number of a Legendrian knot. Received: 17 April 1996 / Revised: 12 May 1999 / Published online: 28 June 2000     

The modified method of characteristics with adjusted advection

Summary. The MMOC procedure for approximating the solutions of transport-dominated diffusion problems does not automatically preserve integral conservation laws, leading to (mass) balance errors in many kinds of flow problems. The variant, called the MMOCAA, discussed herein preserves the conservation law at a minor additional computational cost. It is shown that its solution, in either Galerkin or finite difference form, converges at the same rates as were proved earlier by Dougl as and Russell for the standard MMOC procedure. Received June 25, 1997 / Revised version received October 6, 1998 / Published online: July 7, 1999