Fixing the functoriality of Khovanov homology

dc.contributor.authorClark, David
dc.contributor.authorMorrison, Scott
dc.contributor.authorWalker, Kevin
dc.date.accessioned2015-12-07T22:54:26Z
dc.date.issued2009
dc.date.updated2016-02-24T11:22:59Z
dc.description.abstractWe describe a modification of Khovanov homology [Duke Math. J. 101 (2000) 359-426], in the spirit of Bar-Natan iGeom. Topol. 9 (2005) 1443-1499], which makes the theory properly functorial with respect to link cobordisms. This requires introducing "disorientations" in the category of smoothings and abstract cobordisms between them used in Bar-Natan's definition. Disorientations have "seams" separating oppositely oriented regions, coming with a preferred normal direction. The seams satisfy certain relations (just as the underlying cobordisms satisfy relations such as the neck cutting relation). We construct explicit chain maps for the various Reidemeister moves, then prove that the compositions of chain maps associated to each side of each of Carter, Reiger and Saito's movie moves oJ. Knot Theory Ramifications 2 (1993) 251284; Adv. Math. 127 (1997) 1-51] always agree. These calculations are greatly simplified by following arguments due to Bar-Natan and Khovanov, which ensure that the two compositions must agree, up to a sign. We set up this argument in our context by proving a result about duality in Khovanov homology, generalising previous results about mirror images of knots to a "local" result about tangles. Along the way, we reproduce Jacobsson's sign table eAlgebr. Geom. Topol. 4 (2004) 1211-1251] for the original "unoriented theory", with a few disagreements.
dc.identifier.issn1364-0380
dc.identifier.urihttp://hdl.handle.net/1885/28199
dc.publisherUniversity of Warwick
dc.sourceGeometry and Topology
dc.subjectKeywords: Functoriality; Khovanov homology; Link cobordism
dc.titleFixing the functoriality of Khovanov homology
dc.typeJournal article
local.bibliographicCitation.issue1499-1582
local.contributor.affiliationClark, David, Randolph-Macon College
local.contributor.affiliationMorrison, Scott, College of Physical and Mathematical Sciences, ANU
local.contributor.affiliationWalker, Kevin, Microsoft Station Q
local.contributor.authoremailu5228111@anu.edu.au
local.contributor.authoruidMorrison, Scott, u5228111
local.description.embargo2037-12-31
local.description.notesImported from ARIES
local.identifier.absfor010112 - Topology
local.identifier.absfor010103 - Category Theory, K Theory, Homological Algebra
local.identifier.absseo970101 - Expanding Knowledge in the Mathematical Sciences
local.identifier.ariespublicationu4743872xPUB56
local.identifier.citationvolume13
local.identifier.doi10.2140/gt.2009.13.1499
local.identifier.scopusID2-s2.0-70350001281
local.identifier.thomsonID000264723800008
local.identifier.uidSubmittedByu4743872
local.type.statusPublished Version

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