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- Benjamin Audoux
- ArXiv
- 2013

We use Khovanov homology to define families of LDPC quantum error-correcting codes: unknot codes with asymptotical parameters 32`+1 √ 8π` ; 1; 2 ; unlink codes with asymptotical parameters √ 3 2π`6 ; 2; 2 and (2, `)–torus link codes with asymptotical parameters ~n; 1; dn where dn > √ n 1.62 .

Links in 3–space are usually given by diagrams and isotopies of links by sequences of Reidemeister moves (see e.g. [BZ85]). For relatively oriented links (i.e. up to global orientation reversing), there are exactly two different local (i.e. without regarding the rest of the diagram) Reidemeister moves of type II , shown in Figure 1, and eight local… (More)

- Benjamin Audoux
- 2009

Using the combinatorial description for knot Heegaard–Floer homology, we give a generalization to singular knots that does fit in the general program of categorification of Vassiliev finite–type invariants theory. Introduction Since the categorification of the Jones polynomial by Mikhail Khovanov in 1999 [Kh00], the study of knots and links via homological… (More)

- Benjamin Audoux
- 2009

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. Introduction… (More)

- Benjamin Audoux
- 2008

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.

These notes were written for a serie of lectures on the Rasmussen invariant and the Milnor conjecture, given at Winter Braids IV in February 2014. Introduction A torus knot is a knot in R3 which can be drawn without crossing on the surface of a trivially embedded solid torus. Up to mirror image, non trivial torus knots are classified by pairs {p, q} of… (More)

- Benjamin Audoux, Alain Couvreur
- ArXiv
- 2015

CSS codes are in one-to-one correspondance with length 3 chain complexes. The latter are naturally endowed with a tensor product ⊗ 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 C, we give a criterion which provides a lower bound on the minimum… (More)

- Benjamin Audoux
- 2008

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. Ozsváth and Z. Szabó gave a categorification of Alexander polynomial. Besides their increased abilities for… (More)

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