Rapport sur la théorie classique des noeuds (1ère partie)
Reidemeister-type moves for surfaces in four-dimensional space
We consider smooth knottings of compact (not necessarily orientable) n-dimensional manifolds in (or ), for the cases n=2 or n=3. In a previous paper we have generalized the notion of the Reidemeister moves of classical knot theory. In this paper we examine in more detail the above mentioned dimensions. Examples are given; in particular we examine projections of twist-spun knots. Knot moves are given which demonstrate the triviality of the 1-twist spun trefoil. Another application is a smooth...
Remark on the equisingularity of families of affine plane curves
We give some criteria for the equisingularity of families of affine plane curves.
Rotational Symmetry: Basic Properties and Application to Knot Manifolds.
Secondary invariants for links
Self-Linking Invariants of Embeddings in the Metastable Range.
S-equivalence et congruence de matrices de Seifert: Une conjecture de Trotter.
Singularities and exotic spheres
Singularities, double points, controlled topology and chain duality.
Smooth spheres in IR4 with four critical points are standard.
Some non-trivial PL knots whose complements are homotopy circles
We show that there exist non-trivial piecewise linear (PL) knots with isolated singularities , n ≥ 5, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally flat, and topological locally flat knots, for which it is known that if the complement has the homotopy type of a circle, then the knot is trivial.
Spacefilling knots.
Special positions for surfaces bounded by closed braids.
Stably hyperbolic ...-hermitian forms and doubly sliced knots.
State-sum invariants of knotted curves and surfaces from quandle cohomology.
Submanifolds of codimension two and homology equivalent manifolds
In this paper new methods of studying codimension two embeddings of manifolds are outlined. Results are stated on geometric periodicity of knot cobordism. The group of local knots of a manifold in a 2-plane bundle is introduced and computed, and applied to -close embeddings. General codimension two splitting theorems are discussed, with applications to equivariant knots and knot cobordism. A general existence theorem for P.L. (non-locally flat) embeddings is also given.The methods involve some...
Sur le rôle de la monodromie entière dans la topologie des singularités
Nous considérons l’action de la monodromie sur l’homologie de la fibre de Milnor d’une singularité complexe. Cette action est plus compliquée que prévu : en effet nous montrons que, sur , elle n’est, en général, pas somme directe de modules cycliques. Nous donnons également des exemples prouvant que la monodromie rationnelle ne détermine pas la monodromie entière et que la monodromie entière ne détermine pas la topologie.
Surfaces in 3-space that do not lift to embeddings in 4-space
A necessary and sufficient condition for an immersed surface in 3-space to be lifted to an embedding in 4-space is given in terms of colorings of the preimage of the double point set. Giller's example and two new examples of non-liftable generic surfaces in 3-space are presented. One of these examples has branch points. The other is based on a construction similar to the construction of Giller's example in which the orientation double cover of a surface with odd Euler characteristic is immersed...
Surfaces in the complex projective plane and their mapping class groups.
The 3-cocycles of the Alexander quandles .