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Triplygraded link homology and Hochschild homology of Soergel bimodules
"... 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 KazhdanLusztig theory. Hochschild homology. Let R be a kalgebra, where k is a field, R e = R ⊗k R op be the enveloping algebra ..."
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Cited by 25 (3 self)
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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 KazhdanLusztig theory. Hochschild homology. Let R be a kalgebra, where k is a field, R e = R ⊗k R op be the enveloping algebra of R, and M be an Rbimodule (equivalently, a left R emodule). The functor of Rcoinvariants 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 Rcoinvariants functor is right exact and its ith derived functor takes M to Tor Re i (R, M). The latter space is also denoted HHi(R, M) and called the ith 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 Rbimodule R by projective Rbimodules and tensor the resolution with M:
The chromatic polynomial of fatgraphs and its categorification
 Advances in Mathematics
, 2008
"... 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 ..."
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Cited by 11 (6 self)
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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 torsionfree part of our chromatic homology is independent of the choice of planar embedding of a graph. We extend our construction and categorify the BollobásRiordan 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.
ON THE FIRST GROUP OF THE CHROMATIC COHOMOLOGY OF GRAPHS
, 2006
"... The algebra of truncated polynomials Am = Z[x]/(xm) plays an important role in the theory of Khovanov and KhovanovRozansky 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 comp ..."
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Cited by 3 (1 self)
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The algebra of truncated polynomials Am = Z[x]/(xm) plays an important role in the theory of Khovanov and KhovanovRozansky 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
Reduced chromatic graph cohomology
"... In this paper we give a new characterization of the hvector 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 ..."
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Cited by 1 (0 self)
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In this paper we give a new characterization of the hvector 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.
Homology of torus links
, 2007
"... In this paper we show that there is a cutoff in the Khovanov homology of (2k, 2kn)torus links, namely that the maximal homological degree of nonzero 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 t ..."
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In this paper we show that there is a cutoff in the Khovanov homology of (2k, 2kn)torus links, namely that the maximal homological degree of nonzero 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
Homology of torus links
, 2006
"... In this paper we show that there is a cutoff in the Khovanov homology of (2k, 2kn)torus links, namely that the maximal nonzero 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 ..."
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In this paper we show that there is a cutoff in the Khovanov homology of (2k, 2kn)torus links, namely that the maximal nonzero 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
3.1 The Vanishing Theorem.................... 6
, 2008
"... 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. ..."
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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 KhovanovRozansky sl(m) homology of knots (in particular (2,n) torus knot). We also predict that our work is
3.1 The Vanishing Theorem....................... 5
, 2008
"... 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. ..."
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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 KhovanovRozansky sl(m) homology of knots (in particular (2, n) torus knots). We also predict that our work is connected with Hochschild and Connes cyclic
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, 2006
"... In this paper we prove the knight move theorem for the chromatic graph cohomologies with rational coefficients introduced by L. HelmeGuizon and Y. Rong. Namely, for a connected graph Γ with n vertices the only nontrivial cohomology groups H i,n−i (Γ), H i,n−i−1 (Γ) come in isomorphic pairs: H i,n− ..."
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In this paper we prove the knight move theorem for the chromatic graph cohomologies with rational coefficients introduced by L. HelmeGuizon and Y. Rong. Namely, for a connected graph Γ with n vertices the only nontrivial 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 nonbipartite, 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 deletioncontraction formula for the Poincaré polynomial.