An invariant of link cobordisms from Khovanov homology.

By Magnus Jacobsson. 2002

164 citations recorded by the source corpus.

In (10), Mikhail Khovanov constructed a homology theory for oriented links, whose graded Euler characteristic is the Jones polynomial. He also explained how every link cobordism between two links induces a homomorphism between their homology groups, and he conjectured the in- variance (up to sign) of this homomorphism under ambient isotopy of the link cobordism. In this paper we prove this conjecture, after having made a necessary improvement on its statement. We also introduce polynomial Lefschetz numbers of cobordisms from a link to itself such that the Lef- schetz polynomial of the trivial cobordism is the Jones polynomial. These polynomials can be computed on the chain level. AMS Classification 57Q45; 57M25

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/66e5ee39665d6cc2e3bc364ab20335c7d889c74f

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Citations in the corpus

Knot invariants and higher representation theory II: the categorification of quantum knot invariants (2010, background): “ of this paper, it’s far from clear how they will combine. 3.4. Functoriality. One of the most remarkable properties of Khovanov homology is its functoriality with respect to cobordisms between knots [Jac04]. This property is not only theoretically satisfying but also played an important role in Ras”
Fixing the functoriality of Khovanov homology (2007, methodology): “1 (ie in the original unoriented theory), in Table 2, we see that in most cases we agree with the signs Jacobsson observed [10].”
A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products (2005, methodology): “riﬁcation of all these structures is believed to produce invariants of 2-knots in four dimensions. So far, invariants of link cobordisms were obtained via a categoriﬁcation of the Jones polynomial in [Jac04] and [Kho02]. Their representation theoretic interpretation requires further steps in the “ca”
Functoriality of colored link homologies (2017, methodology): “For N = 2, functorial theories were obtained by Caprau [7], Clark–Morrison–Walker [13] and Blanchet [5] after Khovanov’s original theory turned out to have a sign ambiguity, see Jacobsson [20] and Bar-Natan [2].”
An oriented model for Khovanov homology (2010, methodology): “Functoriality Extension of Khovanov homology to link cobordisms and functoriality up to sign was conjectured by Khovanov and established by Jacobson [8], Khovanov [12] and Bar-Natan [2].”
On Khovanov's cobordism theory for SU3 knot homology (2006, background): “8 The su3 cobordism theory is functorial; in particular the sign problems seen in the su2 case [2, 7, 16] don’t occur.”
Category O and slk link invariants (2007, background): “This natural transformation should become an invariant of 2-tangles as in [Str4], [Khov3], [ Jac ].”
Legendrian knots and exact Lagrangian cobordisms (2012, methodology): “ map Kh(Λi+1) → Kh(Λi). Lemma 9.2. The map Ψ′ L : Kh(Λn) → Kh(Λ1) induced by the decomposable exact Lagrangian cobordism L agrees with ΨL, where we view L as a smooth surface from Λn to Λ1. Proof. In [Ja], the 1-resolution mapKh(K1) → Kh(K) is not used in the definition of the cobordism map ΨL. Inst”
A categorification of finite-dimensional irreducible representations of quantum $${\mathfrak{sl}_2}$$ and their tensor products (2007, methodology): “So far, invariants of link cobordisms were obtained via a categorification of the Jones polynomial in [ Jac04 ] and [Kho06], [Str05], [Str06].”
Transverse knots distinguished by knot Floer homology (2007, background): “(The analogous theorem is true for Khovanov homology [16, 6].”
Quantum link homology via trace functor I (2016, methodology): “movie presentations and it is proven that up to sign JSK does not depend on the presentation chosen [BN05]. Unfortunately, direct computation shows that the sign of JSKdoes depend on the presentation [Jac04]. 11 There are a few approaches to attack the sign issue. In case of F = R2 one can use the d”
A slice genus lower bound from sl(n) Khovanov–Rozansky homology (2007, background): “Furthermore these homology groups provided a categorification of the Jones polynomial P2(D), by which is meant P2(q) = ∑ i,j (−1)iqjdim(HKhi,j2 ) A powerful facet of the HKh2 theory was conjectured by Khovanov (and later proved by Jacobsson [3]), namely that the homology theory should be functorial ”
Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, background): “As originally defined, Khovanov homology is a projective functorial invariant, meaning that to a cobordism f : T → T ′ between two tangles one can assign a map Kh(f) : Kh(T ) → Kh(T ) between the respective homologies well defined only up to a ±1 sign [33, 31, 2, 26].”
On the spectral sequence from Khovanov homology to Heegaard Floer homology (2008, background): “For instance, a cobordism Z ⊂ S3× [0, 1] from L1 to L2 induces a map from K̃h(L1) to K̃h(L2) [12, 11].”
A refinement of Rasmussen's s-invariant (2012, methodology): “ster moves connecting D and D′. Let Φ: Xj Kh(D ′) → Xj Kh(D) denote the stable homotopy equivalence given by [LSa, Theorem 1] and let F : Khi,j(D) → Khi,j(D′) denote the isomorphism given by [Kho02], [Jac04] or [BN05]. Then the following diagram commutes up to sign: Hei(Xj Kh(D)) Φ∗ ∼= // ∼= Hei(Xj”
An introduction to diagrammatic algebra and categorified quantum sl(2) (2011, background): “Khovanov homology is a strictly stronger knot invariant [9], but even more interestingly, it is a functorial knot invariant [32, 28, 17, 11].”