## When the theories meet: Khovanov homology as Hochschild homology of links, arXiv:math.GT/0509334

Citations: | 11 - 3 self |

### BibTeX

@MISC{Przytycki_whenthe,

author = {Jozef H. Przytycki},

title = {When the theories meet: Khovanov homology as Hochschild homology of links, arXiv:math.GT/0509334},

year = {}

}

### OpenURL

### Abstract

ABSTRACT. We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a (2,n)-torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we prove the concrete connection. In the general case of Khovanov-Rozansky, sl(n), homology and their deformations we conjecture the connection. The best framework to explore our ideas is to use a comultiplicationfree version of Khovanov homology for graphs developed by L. Helme-Guizon and Y. Rong and extended here to M-reduced case, and in the case of a polygon to noncommutative algebras. In this framework we prove that for any unital algebra A the Hochschild homology of A is isomorphic to graph homology over A of a polygon. We expect that this

### Citations

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Citation Context ... and, in some cases, links (with the classical Khovanov homology as the main example). In examples of functors from “supersets” we utilize comultiplication in the underlying algebra A, after Khovanov =-=[Kh-1]-=-, we assume that A is a Frobenius algebra [Abr, Kock]. In the third section we prove our main result relating Hochschild homology HH∗(A) to graph cohomology and Khovanov homology of links, and the hom... |

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Citation Context ...is a free ( n) m k-module with a basis vi1 ∧vi2 ∧...∧vim, i1 < i2 < ... < im. The above theorem is a special case of Hochschild-Konstant-Rosenberg theorem about Hochschild homology of smooth algebras =-=[HKR]-=-, which we discuss in section 4. Here we stress, after Loday, that the isomorphism ε∗ : S(V ) ⊗ Λ ∗ V → HH∗(S(V )) is induced by a chain map, that is not true in general for smooth algebras. S(V )⊗Λ ∗... |

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Citation Context ...m(g−1) + tq mg ). Assume that m = 2 in Corollary 4.3. Then, using Theorem 2.7 we can recover Khovanov computation of homology of the torus link T2,n [Kh-1, Kh-2]. In particular we get: Corollary 4.4. =-=[Kh-2]-=- Let T2,−n be a left-handed torus link of type (2, −n), n > 2. Then the torsion part of the Khovanov homology of T2,−n is given14 When the theories meet by (in the description of homology we use nota... |

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Citation Context ... more delicate (co)homology theory for ribbon graphs (flat vertex graph) using the fact that they embed uniquely into the closed surface. For A = A2 it can be achieved using the approach presented in =-=[APS]-=-, while for more general (Frobenius) algebras it is not yet done (most likely one should not use Frobenius algebra alone but its proper enhancement like in A2 case). In [H-P-R] we proved the following... |

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Citation Context ...8(T2,−n) = Hn−6,3n−12(T2,−n) = ... = H−n+4,−n+8(T2,−n) = Z2. For a right-handed torus link of type (2,n), n > 2, we can use the formula for the mirror image (Khovanov duality theorem; see for example =-=[A-P]-=-,A-P-S): H−i,−j( ¯ D) = Hij(D)/Tor(Hij(D)) ⊕ Tor(Hi−2,j(D)). The result on Hochschild homology of symmetric algebras has a major generalization to the large class of algebras called smooth algebras. T... |

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Citation Context ...tion σ ∈ Sn acts by σ((a0,a1,...,an)) = (a0,a σ −1 (1),...,a σ −1 (n)). In the second section we describe Khovanov homology of links and its (“comultiplication-free”) version for graphs introduced in =-=[H-R-2]-=-. In order to compare them with Hochschild homology we offer various generalizations, relaxing a condition that underlying algebra needs to be commutative (in the case of a polygon or a line graph), a... |

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Citation Context ...Definition 2.2 and Remark 2.3. In order to define Khovanov cohomology on any link diagram (and take both multiplication and comultiplication into account) we have to define 3 We are motivated here by =-=[Sto]-=-.Khovanov homology as Hochschild homology of links 7 cohomology on any signed planar graph. We can start, as in Definition 2.1, from the very general setting (again cohomology of a functor) and to pr... |

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Citation Context ... this setting we recall and generalize the concept of graph cohomology [H-R-2] and of classical Khovanov homology for unoriented framed links. We follow, in part, the exposition in [H-R-1, H-R-2] and =-=[Vi]-=-. We review, after [H-P-R], the connection between Khovanov homology of links and graph cohomology of associated Tait graphs. We define homology of link diagrams related to graph cohomology for any co... |

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Citation Context ... a polynomial in Z[x]. We discuss the general case later, here let us notice that two special cases of p(x) = x m and p(x) = x m − 1 are of great interest in knot theory (in KhovanovRozansky homology =-=[Kh-R-1]-=- and its deformations [Gor]). Let us apply first the knowledge of Hochschild homology for Am = Z[x]/(x m ) (compare [Lo]) and solving Conjectures 30 and 31 of [H-P-R]. Theorem 4.2. (Free) The Poincaré... |

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1 |
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Citation Context ...uss the general case later, here let us notice that two special cases of p(x) = x m and p(x) = x m − 1 are of great interest in knot theory (in KhovanovRozansky homology [Kh-R-1] and its deformations =-=[Gor]-=-). Let us apply first the knowledge of Hochschild homology for Am = Z[x]/(x m ) (compare [Lo]) and solving Conjectures 30 and 31 of [H-P-R]. Theorem 4.2. (Free) The Poincaré polynomial of HH∗∗(Am) is ... |

1 |
Torsion in Graph Homology, for Fundamenta Mathematicae, preprint
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(Show Context)
Citation Context ...tric tensor algebras. More calculations are reviewed in Section 4 in which we use our main result, Theorem 3.1, to obtain new results in Khovanov homology, in particular solving some conjectures from =-=[H-P-R]-=-. We follow [Lo] in our exposition of Hochschild homology. Let k be a commutative ring and A a k-algebra (not necessarily commutative). Let M be a bimodule over A that is a k-module on which A operate... |

1 |
Personal communication at AMS-IMS-SIAM
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(Show Context)
Citation Context ...heorem 2.7, relation between classical Khovanov homology of (2,n + 1) torus link and Hochschild homology of A2 = Z[x]/(x 2 ), follows. This relation was also observed independently by Magnus Jacobson =-=[Jac]-=-. In this paper we prove more general result. In order to formulate it we use an extended version of (Khovanov type) graph cohomology (working with noncommutative algebras and M-reduced cohomology): (... |

1 |
and Harrison cohomology of complete intersections, Appendix to the paper Quantization on Curves by C.Fronsdal, e-print: http://front.math.ucdavis.edu/math-ph/0507021
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Citation Context ...mooth algebra. For example, C[x,y])(x 2 y 3 ) or Z[x]/(x m ) are not smooth. The broadest, to my knowledge, treatment of Hochschild homology of algebras C[x1,...,xn]/(Ideal) is given by Kontsevich in =-=[Kon]-=-. For us the motivation came from one variable polynomials, Theorem 40 of [H-P-R]. In particular we generalize Theorem 40(i) from a triangle to any polygon that is we compute the graph cohomology of a... |

1 | Torsion of the Khovanov Homology, Geometry and Topology - Shumakovitch |

1 |
forms on regular affine algebras
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- 1962
(Show Context)
Citation Context ...s a free ( ) n m k-module with a basis vi1 ∧vi2 ∧...∧vim, i1 < i2 < ... < im. The above theorem is a special case of Hochschild-Konstant-Rosenberg theorem about Hochschild homology of smooth algebras =-=[HKR]-=-, which we discuss in section 4. Here we stress, after Loday, that the isomorphism ε∗ : S(V ) ⊗ Λ ∗ V → HH∗(S(V )) is induced by a chain map, that is not true in general for smooth algebras. S(V )⊗Λ ∗... |

1 |
When the theories meet
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(Show Context)
Citation Context ...s. More calculations are reviewed in Section 4 in which we use our main result, Theorem 3.1, to obtain new results in Khovanov homology, in particular solving some conjectures from [H-P-R]. We follow =-=[Lo]-=- in our exposition of Hochschild homology. Let k be a commutative ring and A a k-algebra (not necessarily commutative). Let M be a bimodule over A that is a k-module on which A operates linearly on th... |