Links associated with generic immersions of graphs.
Löbell manifolds revised.
Locally unknotted spines of Heegaard splittings.
Longitude Floer homology and the Whitehead double.
Mapping class group and the Casson invariant
Using a new definition of the second and third Johsnon homomorphisms, we simplify and extend the work of Morita on the Casson invariant of homology-spheres defined by Heegard splittings. In particular, we calculate the Casson invariant of the homology-sphere obtained by gluing two handlebodies along a homeomorphism of the boundary belonging to the Torelli subgroup.
Matrix factorizations and link homology
For each positive integer n the HOMFLYPT polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links with this polynomial as the Euler characteristic. The core of our construction utilizes the theory of matrix factorizations, which provide a linear algebra description of maximal Cohen-Macaulay modules on isolated hypersurface singularities.
Maximal Thurston-Bennequin number of two-bridge links.
Minimal surface representations of virtual knots and links.
Mirror-curve Codes for Knots and Links
Monoids of linking pairings and orthogonal summands. (Monoide des enlacements et facteurs orthogonaux.)
Mp-small summands increase knot width.
New categorifications of the chromatic and dichromatic polynomials for graphs
For each graph G, we define a chain complex of graded modules over the ring of polynomials whose graded Euler characteristic is equal to the chromatic polynomial of G. Furthermore, we define a chain complex of doubly-graded modules whose (doubly) graded Euler characteristic is equal to the dichromatic polynomial of G. Both constructions use Koszul complexes, and are similar to the new Khovanov-Rozansky categorifications of the HOMFLYPT polynomial. We also give a simplified definition of this triply-graded...
New topologically slice knots.
Nielsen number is a knot invariant
We show that the Nielsen number is a knot invariant via representation variety.
Non abelian Reidemeister torsion and volume form on the SU(2)-representation space of knot groups
For a knot in the 3-sphere and a regular representation of its group into SU(2) we construct a non abelian Reidemeister torsion form on the first twisted cohomology group of the knot exterior. This non abelian Reidemeister torsion form provides a volume form on the SU(2)-representation space of . In another way, we construct using Casson’s original construction a natural volume form on the SU(2)-representation space of . Next, we compare these two apparently different points of view on the representation...
Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers
Let be a dg algebra over and let be a dg -bimodule. We show that under certain technical hypotheses on , a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the Hochschild homology of the derived tensor product and converges to the Hochschild homology of . We apply this result to bordered Heegaard Floer theory, giving spectral sequences associated to Heegaard Floer homology groups of certain branched and unbranched double covers.
Noncommutative knot theory.
Non-slice linear combinations of algebraic knots
We show that the subgroup of the knot concordance group generated by links of isolated complex singularities intersects the subgroup of algebraically slice knots in an infinite rank subgroup.
Non-triviality of the -polynomial for knots in .
On 3-manifold invariants arising from finite-dimensional Hopf algebras.