Benjamin Audoux

A Jones polynomial for braid-like isotopies of oriented links and its categorification

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. AMS Classification 57M27; 20F36

Singular link Floer homology

We define a grid presentation for singular links, ie 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. 57M25

On Usual, Virtual and Welded knotted objects up to homotopy

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 the techniques of Gauss diagram formulae.

An application of Khovanov homology to quantum codes

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.

Khovanov homology and star-like isotopies

Abstract A star-like isotopy for oriented links in 3-space is an isotopy which uses only Reidemeister moves which correspond to the following singularities of planar curves: ,  ,  ,  . We define a link polynomial derived from the Jones polynomial which is, in general, only invariant under star-like isotopies and we categorify it.

Extensions of some classical local moves on knot diagrams

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. Interrelationship between these moves is discussed and, for each of these move, we provide an algebraic classification. We address the question of relevant welded extensions for classical moves in the sense that the classical quotient of classical object embeds into the welded quotient of welded objects. As a by-product, we obtain that all of the above local moves are unknotting operations for welded (long) knots. We also mention some topological interpretations for these combinatorial quotients.

On codimension two embeddings up to link‐homotopy

We consider knotted annuli in 4–space, called 2–string-links, which are knotted surfaces in codi-mension 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 link-homotopic to a ribbon one; this allows to use the homotopy classification obtained in the ribbon case by P. Bellingeri and the authors. Along the way, we give a Roseman-type result for immersed surfaces in 4–space. We also discuss the case of ribbon k–string links, for k ≥ 3.

Surfaces with pulleys and Khovanov homology

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

On multiplicity of eigenvalues and symmetry of eigenfunctions of the

We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the p-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains Ω ⊂ R^N. By means of topological arguments, we show how symmetries of Ω help to construct subsets of W_0^(1,p)(Ω) with suitably high Krasnoselski˘ i genus. In particular, if Ω is a ball B ⊂ R^N , we obtain the following chain of inequalities: λ_2(p; B) ≤ · · · ≤ λ_(N+1)(p; B) ≤ λ_eq(p; B). Here λ_i(p; B) are variational eigenvalues of the p-Laplacian on B, and λ_eq(p; B) is the eigenvalue which has an associated eigenfunction whose nodal set is an equatorial section of B. If λ_2(p; B) = λ(p; B), as it holds true for p = 2, the result implies that the multiplicity of the second eigenvalue is at least N. In the case N = 2, we can deduce that any third eigenfunction of the p-Laplacian on a disc is nonradial. The case of other symmetric domains and the limit cases p = 1, p = ∞ are also considered.