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We use Khovanov homology to define families of LDPC quantum error-correcting codes: unknot codes
A braid-like isotopy for links in 3-space is an isotopy which uses only those Reidemeister moves which occure 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.
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.
Using the combinatorial description for knot Heegaard–Floer ho-mology, we give a generalization to singular knots that does fit in the general program of categorification of Vassiliev finite–type invariants theory.
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)
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)