Benjamin Audoux

Aix-Marseille Université | AMU · Département de mathématiques

The reduced peripheral system was introduced by Milnor [18] in the 1950s for the study of links up to link-homotopy, that is, up to homotopies leaving distinct components disjoint; this invariant, however, fails to classify links up to link-homotopy for links of four or more components. The purpose of this paper is to show that the topological info...

We define numerical link-homotopy invariants of link maps of any number of components, which naturally generalize the Kirk invariant. The Kirk invariant is a link-homotopy invariant of 2-component link maps given by linking numbers of loops based at self-singularities of each component with the other spherical component; our invariants use instead...

We develop a general diagrammatic theory of welded graphs, and provide an extension of Satoh's Tube map from welded graphs to ribbon surface-links. As a topological application, we obtain a complete link-homotopy classification of so-called knotted punctured spheres in 4-space, by means of the 4-dimensional Milnor invariants introduced by the autho...

We construct a universal finite type invariant for knots in homology 3-spheres, refining Kricker's lift of the Kontsevich integral. This provides a full diagrammatic description of the graded space of finite type invariants of knots in homology 3-spheres.

We characterize, in an algebraic and in a diagrammatic way, Milnor string link invariants indexed by sequences where any index appears at most $k$ times, for any fixed $k\ge 1$. The algebraic characterization is given in terms of an Artin-like action on the so-called $k$-reduced free groups; the diagrammatic characterization uses the langage of wel...

We generalize Milnor link invariants to all types of surface-links in 4-space (possibly with boundary) and more generally to all codimension 2 embeddings. This is achieved by using the notion of cut-diagram, which is a higher dimensional generalization of Gauss diagrams, associated to codimension 2 embeddings in Euclidian spaces. We define a notion...

The reduced peripheral system was introduced by Milnor in the fifties for the study of links up to link-homotopy, i.e. up to isotopies and crossing changes within each link component. However, for four or more components, this invariant does not yield a complete link-homotopy invariant. This paper provides two characterizations of links having the...

In the setting of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries, there are two candidates to be universal invariants, defined, respectively, by Kricker and Lescop. In a previous paper, the second author defined maps between spaces of Jacobi diagrams. Injectivity...

In the setting of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries, there are two candidates to be universal invariants, defined respectively by Kricker and Lescop. In a previous paper, the second author defined maps between spaces of Jacobi diagrams. Injectivity fo...

We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the $p$-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains $\Omega \subset \mathbb{R}^N$. By means of topological arguments, we show how symmetries of $\Omega$ help to construct subsets of $W_0^{1,p}(\Omega)$ with suit...

We consider knotted annuli in 4-space, called 2-string-links, which are knotted surfaces in codimension two that are naturally related, via closure operations, to both 2-links and 2-torus links. We classify 2-string-links up to link-homotopy by means of a 4-dimensional version of Milnor invariants. The key to our proof is that any 2-string link is...

Baker showed that 10 of the 12 classes of Berge knots are obtained by surgery on the minimally twisted 5-chain link. In this article we enumerate all hyperbolic knots in S^3 obtained by surgery on the minimally twisted 5 chain link that realize the maximal known distances between slopes corresponding to exceptional (lens, lens), (lens, toroidal), (...

In the present paper, we consider local moves on classical and welded diagrams: (self-)crossing change, (self-)virtualization, virtual conjugation, Delta, fused, band-pass and welded band-pass moves. Interre- lationship between these moves is discussed and, for each of these move, we provide an algebraic classification. We address the question of r...

CSS codes are in one-to-one correspondance with length 3 chain complexes. The latter are naturally endowed with a tensor product $\otimes$ which induces a similar operation on the former. We investigate this operation, and in particular its behavior with regard to minimum distances. Given a CSS code $\mathcal{C}$, we give a criterion which provides...

We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or self-virtualizations. We provide a number of results which point out the differences between these various notions. The proofs are mainly based on t...

This note investigates the so-called Tube map which connects welded knots, that is a quotient of the virtual knot theory, to ribbon torus-knots, that is a restricted notion of fillable knotted tori in the 4-sphere. It emphasizes the fact that ribbon torus-knots with a given filling are in one-to-one correspondence with welded knots before quotient...

Ribbon 2-knotted objects are locally flat embeddings of surfaces in 4-space which bound immersed 3-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper, we consider ribbon tubes, which are knotted annuli bounding ribbon 3-balls....

We use Khovanov homology to define families of LDPC quantum error-correcting codes: unknot codes with asymptotical parameters [[3^(2l+1)/sqrt(8{\pi}l);1;2^l]]; unlink codes with asymptotical parameters [[sqrt(2/2{\pi}l)6^l;2^l;2^l]] and (2,l)-torus link codes with asymptotical parameters [[n;1;d_n]] where d_n>\sqrt(n)/1.62.

In this paper, we define surfaces with pulleys which are unions of 1- and 2-dimensional manifolds, glued together on a finite number of /3-labeled points of their interiors. Then, by seeing them as cobordisms, we give a refinement of Bar-Natan's geometrical construction of Khovanov homology which can be applied to different notions of refined links...

A star-like isotopy for oriented links in 3-space is an isotopy which uses only Reidemeister II moves with opposite orientations and Reidemeister III moves with alternating orientations when checking the strands clockwise (or anticlockwise). We define a link polynomial derived from the Jones polynomial which is, in general, only invariant under sta...

A categorification of a polynomial link invariant is an homological invariant which contains the polynomial one as its graded Euler characteristic. This field has been initiated by Khovanov categorification of the Jones polynomial. Later, P. Ozsvath and Z. Szabo gave a categorification of Alexander polynomial. Besides their increased abilities for...

La catégorification d'un invariant polynomial d'entrelacs I est un invariant de type homologique dont la caractéristique d'Euler gradue est égale à I. On pourra citer la catégorification originelle du polynôme de Jones par M. Khovanov ou celle du polynôme d'Alexander par P. Ozsvath et Z. Szabo. Outre leur capacité accrue à distinguer les noeuds, ce...

We define a grid presentation for singular links i.e. links with a finite number of rigid transverse double points. Then we use it to generalize link Floer homology to singular links. Besides the consistency of its definition, we prove that this homology is acyclic under some conditions which naturally make its Euler characteristic vanish.

A braid-like isotopy for links in 3-space is an isotopy which uses only those Reidemeister moves which occur in isotopies of braids. We define a refined Jones polynomial and its corresponding Khovanov homology which are, in general, only invariant under braid-like isotopies.

We give a geometrical construction for singular link Floer homology HFV, then we use it to prove that it vanishes for any singular connected sum of links. Link Floer homology is an invariant of link in closed 3–manifolds which categorify the Alexander poly-nomial in the sense that the latter is recovered from the former as its graded Euler charaste...