On the welded Tube map

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 under classical Reidemeister moves and reformulates these moves and the known sources of non-injectivity of the Tube map in terms of filling changes.Comment: 23 pages ; v2: an error corrected and stylistic modifications ; to appear in Contemporary Mathematic

Singular link Floer homology

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.Comment: 29 pages, many figure

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.Comment: 20 page

Characterization of the reduced peripheral system of links

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 same reduced peripheral system: a diagrammatical one, in terms of link diagrams, seen as welded diagrams up to self-virtualization, and a topological one, in terms of ribbon solid tori in 4--space up to ribbon link-homotopy.Comment: 12 pages ; v2:names and surnames of the authors have been put in the right order, v3:an alternative proof for Lemma 1.16 has been given, v4:minor stylistic change

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.Comment: 19 pages, many figure

On hyperbolic knots in S^3 with exceptional surgeries at maximal distance

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), (lens, Seifert fibred spaces) pairs. In light of Baker's work, the classification in this paper conjecturally accounts for 'most' hyperbolic knots in S^3 realizing the maximal distance between these exceptional pairs. All examples obtained in our classification are realized by filling the magic manifold. The classification highlights additional examples not mentioned in Martelli and Petronio's survey of the exceptional fillings on the magic manifold. Of particular interest, is an example of a knot with two lens space surgeries that is not obtained by filling the Berge manifold.Comment: 30 pages, 5 figures. This revised version has some improvements in the exposition. The main theorems remain as in the last versio

Rasmussen invariant and Milnor conjecture

International audienceThese notes were written for a serie of lectures on the Rasmussen invariant and the Milnor conjecture, given at Winter Braid IV on february 2014

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.Comment: 14 pages. This paper is an expanded version of a former section, now removed (section 5 in versions 1 and 2) of arXiv:1407.0184. To appear in Journal of the Mathematical Society of Japa

Homotopy classification of ribbon tubes and welded string links

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 and ribbon torus-links, which are natural analogues of string links and links, respectively. We show how ribbon tubes naturally act on the reduced free group, and how this action classifies ribbon tubes up to link-homotopy, that is when allowing each component to cross itself. At the combinatorial level, this provides a classification of welded string links up to self-virtualization. This generalizes a result of Habegger and Lin on usual string links, and the above-mentioned action on the reduced free group can be refined to a general "virtual extension" of Milnor invariants. As an application, we obtain a classification of ribbon torus-links up to link-homotopy.Comment: 33p. ; v2: typos and minor corrections ; v3: Introduction rewritten, exposition revised, references added. Section 5 of the previous version was significantly expanded and was separated into another paper (arXiv:1507.00202) ; v4: typos and minor corrections ; to appear in Annali della scuola Normale Superiore de Pisa (classe de scienze