We trade matrix factorizations and Koszul complexes for Hochschild homology of Soergel bimodules to modify the construction of triplygraded link homology and relate it to Kazhdan-Lusztig theory. Hochschild homology. Let R be a k-algebra, where k is a field, R e = R ⊗k R op be the enveloping algebra of R, and M be an R-bimodule (equivalently, a left R e-module). The functor of R-coinvariants associates to M the factorspace MR = M/[R, M], where [R, M] is the subspace of M spanned by vectors of the form rm − mr. We have MR = R ⊗Re M. The R-coinvariants functor is right exact and its i-th derived functor takes M to Tor Re i (R, M). The latter space is also denoted HHi(R, M) and called the i-th Hochschild homology of M. The Hochschild homology of M is the direct sum HH(R, M) def = ⊕ HHi(R, M). i≥0 To compute Hochschild homology, we choose a resolution of the R-bimodule R by projective R-bimodules and tensor the resolution with M:

Abstract. Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov homology of an associated link. We apply this connection with Khovanov homology to show that the torsion-free part of our chromatic homology is independent of the choice of planar embedding of a graph. We extend our construction and categorify the Bollobás-Riordan polynomial (a generalisation of the Tutte polynomial to embedded graphs). We prove that both our chromatic homology and the Khovanov homology of an associated link can be recovered from this categorification. 1.

The algebra of truncated polynomials Am = Z[x]/(xm) plays an important role in the theory of Khovanov and Khovanov-Rozansky homology of links. We have demonstrated that Hochschild homology is closely related to Khovanov homology via comultiplication free graph cohomology. It is not difficult to compute Hochschild homology of Am and the only torsion, equal to Zm, appears in gradings (i, m(i+1) 2) for any positive odd i. We analyze here the grading of graph cohomology which is producing torsion for a polygon. We find completely the cohomology H 1,v−1 (G) and H

In this paper we give a new characterization of the h-vector of the chromatic polynomial of a graph, i.e. the vector (h0,..., hn) of coefficients of the chromatic polynomial pΓ(λ) = h0λ(λ − 1) n−1 − h1λ(λ − 1) n−2 + · · · + (−1) n−1 hn−1λ. We introduce reduced chromatic cohomology of a graph and show that hi are its Betti numbers. We then discuss various combinatorial properties of these cohomologies.

In this paper we show that there is a cut-off in the Khovanov homology of (2k, 2kn)-torus links, namely that the maximal homological degree of non-zero homology group of (2k, 2kn)-torus link is 2k 2 n. Furthermore, we calculate explicitely the homology groups in homological degree 2k 2 n and prove that it coincides with the centre of the ring H k of crossingless matchings, introduced by M. Khovanov in [7]. Also we give an explicit formula for the ranks of the homology groups of (3, n)-torus knots for every n ∈ N. 1

In this paper we show that there is a cut-off in the Khovanov homology of (2k, 2kn)-torus links, namely that the maximal non-zero homology group of (2k, 2kn)-torus link is in homological degree 2k 2 n. Furthermore, we calculate explicitly the homology group in homological degree 2k 2 n and show that it coincides with the centre of the ring H k introduced by M. Khovanov in [8]. Also we give an explicit formula for the ranks of the homology groups of (3, n)-torus knots for every n ∈ N. 1

Khovanov homology for knots has generated a flurry of activity in the topology community. This paper studies the Khovanov type cohomology for graphs with a special attention to torsion. When the underlying algebra is Z[x]/(x 2), we determine precisely those graphs whose cohomology contains torsion. For a large class of algebras, we show that torsion often occurs. Our investigation of torsion led to other related general results. Our computation could potentially be used to predict the Khovanov-Rozansky sl(m) homology of knots (in particular (2,n) torus knot). We also predict that our work is

Khovanov homology for knots has generated a flurry of activity in the topology community. This paper studies the Khovanov type cohomology for graphs with a special attention to torsions. When the underlying algebra is Z[x]/(x 2), we determine precisely those graphs whose cohomology contains torsion. For a larger class of algebras, we show that torsion often occurs. Our investigation of torsion led to other related general results. The ideas of this paper could potentially be used to predict the Khovanov-Rozansky sl(m) homology of knots (in particular (2, n) torus knots). We also predict that our work is connected with Hochschild and Connes cyclic

In this paper we prove the knight move theorem for the chromatic graph cohomologies with rational coefficients introduced by L. Helme-Guizon and Y. Rong. Namely, for a connected graph Γ with n vertices the only non-trivial cohomology groups H i,n−i (Γ), H i,n−i−1 (Γ) come in isomorphic pairs: H i,n−i (Γ) ∼ = H i+1,n−i−2 (Γ) for i � 0 if Γ is non-bipartite, and for i> 0 if Γ is bipartite. As a corollary, the ranks of the cohomology groups are determined by the chromatic polynomial. At the end, we give an explicit formula for the Poincaré polynomial in terms of the chromatic polynomial and a deletion-contraction formula for the Poincaré polynomial.

Nesta tese, trabalhamos com a homologia de Khovanov de enlaces e com a homologia de grafos. A homologia de Khovanov de enlaces consta dum complexo graduado de cadeias que é um invariante de enlace, cuja característica de Euler é igual ao polinómio de Jones do enlace, e portanto pode ser vista como a “categorificação ” do polinómio de Jones. Provamos que o primeiro grupo de homologia dum nó de trança positiva (positive braid knot) é trivial. Depois, provamos que os nós de toro, não alternados são homologicamente espessos (thick). Também, demonstramos que podemos diminuir o número de torções completas de nós de toro sem mudar a homologia de baixo grau, implicando a existência de homologia estável de nós de toro. Mais, provamos a maior parte das propriedades anteriores também para a homologia de Khovanov-Rozansky. Na homologia de grafos, categorificamos o polinómio dicromático (e em consequência, o polinómio de Tutte) para grafos, através da categorificação dum conjunto infinito de especializações duma variável. Também, categorificamos