Link homology and Frobenius extensions

By Mikhail Khovanov. 2004

155 citations recorded by the source corpus.

Author(s): Khovanov, Mikhail | Abstract: We explain how rank two Frobenius extensions of commutative rings lead to link homology theories and discuss relations between these theories, Bar-Natan theories, equivariant cohomology and the Rasmussen invariant.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/401bd1e3c1b75b8912f1de078e76fe0396b8254a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Matrix factorizations and link homology II (2005, background): “n/–equivariant cohomology ring of CP: A certain version of Bar-Natan link homology [3] from Khovanov [7] should correspond to the potential x(3)C ax: For a common generalization of the U.”
sl.N/-link homology (N 4) using foams and the Kapustin-Li formula (2007, unknown): “We use Khovanov’s idea of “twist equivalence” in [4].”
Deformations of colored sl(N) link homologies via foams (2015, background): “Recall from [QR] that a natural setting for a combinatorial formulation for KhovanovRozansky’s slN link homology is the 2-category NFoam.”
Functoriality of colored link homologies (2017, background): “The first equivariant, or universal, link homology can be traced back to Bar-Natan [2] and Khovanov [25] and it encapsulated both Khovanov homology and Lee’s deformation.”
Categorification of the Jones-Wenzl Projectors (2010, background): “When ̨ D 0 this is the Frobenius algebra which appears in Khovanov’s original construction; see [18].”
The universal sl3–link homology (2006, background): “We refer to the latter as the sl2 –link homologies over C, because they are all deformations of Khovanov’s original link homology whose Euler characteristic equals the Jones polynomial which is well known to be related to the Lie algebra sl2 [8].”
A remark on rasmussen's invariant of knots (2005, methodology): “These theories are obtained from Khovanov’s theory A5 in [3] by specialisation of the variableh andt to elements ofK.”
Gauge theory and Rasmussen's invariant (2011, background): “An alternative viewpoint on the same construction is given in [4].”
Homological algebra of knots and BPS states (2011, background): “ntials. Namely, in doubly-graded knot homologies constructed from matrix factorizations diﬀerentials that relate diﬀerent theories are in one-to-one correspondence with deformations of the potentials [51, 52, 53, 54, 55, 6]: (3.12) diﬀerentials on Hg,R ⇐⇒ deformations of W g,R For example, deformati”
Man and machine thinking about the smooth 4-dimensional Poincaré conjecture (2009, background): “Although this theorem hasn’t appeared in the literature in this form, it follows from the discussion at the end of [28], or by thinking about the invariant of a cut-open knot in the variation of Bar-Natan’s formalism [6] for which the genus 3 surface is a formal parameter.”