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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...
We show that the Nielsen number is a knot invariant via representation variety.
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...
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.
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.
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