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New categorifications of the chromatic and dichromatic polynomials for graphs

Marko Stošić (2006)

Fundamenta Mathematicae

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.

Friedl, Stefan, Teichner, Peter (2005)

Geometry & Topology

Nielsen number is a knot invariant

Alexander Fel'shtyn (2007)

Banach Center Publications

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

Jérôme Dubois (2005)

Annales de l’institut Fourier

For a knot $K$ in the 3-sphere and a regular representation of its group $G_{K}$ 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 $G_{K}$ . In another way, we construct using Casson’s original construction a natural volume form on the SU(2)-representation space of $G_{K}$ . 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

Robert Lipshitz, David Treumann (2016)

Journal of the European Mathematical Society

Let $A$ be a dg algebra over $𝔽_{2}$ and let $M$ be a dg $A$ -bimodule. We show that under certain technical hypotheses on $A$ , a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the Hochschild homology of the derived tensor product $M \otimes_{A}^{L} M$ and converges to the Hochschild homology of $M$ . 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.

Cochran, Tim D. (2004)

Algebraic & Geometric Topology

Non-slice linear combinations of algebraic knots

Matthew Hedden, Paul Kirk, Charles Livingston (2012)

Journal of the European Mathematical Society

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 $A$ -polynomial for knots in $S^{3}$ .

Dunfield, Nathan M., Garoufalidis, Stavros (2004)

Algebraic & Geometric Topology

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