On Khovanov's cobordism theory for SU3 knot homology

By S. Morrison, Ari Nieh. 2006

53 citations recorded by the source corpus.

We reconsider the link homology theory defined by Knovanov in [9] and generalized by Mackaay and Vaz in [15]. With some slight modifications, we describe the theory as a map from the planar algebra of tangles to a planar algebra of complexes of "cobordisms with seams" (actually, a "canopolis"), making it local in the sense of Bar-Natan's local theory of [2]. We show that this "seamed cobordism canopolis" decategorifies to give precisely what you had both hope for and expect: Kuperberg's spider defined in [14]. We conjecture an answer to an even more interesting question about the decategorification of the Karoubi envelope of our cobordism theory. Finally, we describe how the theory is actually completely computable, and give a detailed calculation of the homology of the (2,n) torus knots.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5324f5f0d7ea47868a683881457c03470f126932

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A diagrammatic category for the representation theory of ( Uqsln ) (2007, background): “explicitly local in later work by Ari Nieh and myself [ 25 ].”
The $\mathfrak{sl}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality (2014, background): “5) = • − • is a relation in the Hom-category Hom(1, 1); see [45] or [35] for the full collection of relations.”
Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, methodology): “Renormalizing this invariant via JτKr3 = q2w(τ)JτK3 gives an invariant independent of framing which is (essentially) the same as Morrison-Nieh’s [58] extension of Khovanov’s sl3 link homology [32] to tangles, after setting the 3-, 4-, and 5-dotted spheres equal to zero.”
Categorification of the Jones-Wenzl Projectors (2010, methodology): “We will make frequent use of the following standard lemma in this paper; see [4] and [22].”
Man and machine thinking about the smooth 4-dimensional Poincaré conjecture (2009, background): “Khovanov homology (or its variants categorifying the sl3 polynomial [27, 33, 35] or sln polynomials [29, 30, 43]) may hold promise for establishing that a link L is not slice.”