Fixing the functoriality of Khovanov homology

Clark, David¹ ; Morrison, Scott² ; Walker, Kevin²

¹ Department of Mathematics, Randolph-Macon College, Ashland, VA 23005, USA
² Microsoft Station Q, University of California, Santa Barbara 93106-6105, USA

Geometry & topology, Tome 13 (2009) no. 3, pp. 1499-1582.

Résumé

We describe a modification of Khovanov homology [Duke Math. J. 101 (2000) 359-426], in the spirit of Bar-Natan [Geom. 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 [J. Knot Theory Ramifications 2 (1993) 251-284; 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 [Algebr. Geom. Topol. 4 (2004) 1211-1251] for the original "unoriented theory", with a few disagreements.

Détail

DOI : 10.2140/gt.2009.13.1499

Keywords: Khovanov homology, functoriality, link cobordism

Bibliographie

[1] D Bar-Natan, Khovanov homology for knots and links with up to 11 crossings, from: "Advances in topological quantum field theory" (editor J M Bryden), NATO Sci. Ser. II Math. Phys. Chem. 179, Kluwer Acad. Publ. (2004) 167

[2] D Bar-Natan, Khovanov's homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443

[3] D Bar-Natan, Fast Khovanov homology computations, J. Knot Theory Ramifications 16 (2007) 243

[4] J W Barrett, B W Westbury, Spherical categories, Adv. Math. 143 (1999) 357

[5] D Bisch, Bimodules, higher relative commutants and the fusion algebra associated to a subfactor, from: "Operator algebras and their applications (Waterloo, ON, 1994/1995)" (editors P A Fillmore, J A Mingo), Fields Inst. Commun. 13, Amer. Math. Soc. (1997) 13

[6] C L Caprau, sl(2) tangle homology with a parameter and singular cobordisms, Algebr. Geom. Topol. 8 (2008) 729

[7] J S Carter, J H Rieger, M Saito, A combinatorial description of knotted surfaces and their isotopies, Adv. Math. 127 (1997) 1

[8] J S Carter, M Saito, Reidemeister moves for surface isotopies and their interpretation as moves to movies, J. Knot Theory Ramifications 2 (1993) 251

[9] S I Gelfand, Y I Manin, Methods of homological algebra, Springer (1996)

[10] M Jacobsson, An invariant of link cobordisms from Khovanov homology, Algebr. Geom. Topol. 4 (2004) 1211

[11] V F R Jones, Planar algebras, I

[12] A Joyal, R Street, The geometry of tensor calculus. I, Adv. Math. 88 (1991) 55

[13] M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359

[14] M Khovanov, A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002) 665

[15] M Khovanov, Categorifications of the colored Jones polynomial, J. Knot Theory Ramifications 14 (2005) 111

[16] M Khovanov, L Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008) 1

[17] M Khovanov, L Rozansky, Matrix factorizations and link homology. II, Geom. Topol. 12 (2008) 1387

[18] R Kirby, P Melvin, The $3$–manifold invariants of Witten and Reshetikhin–Turaev for $\mathrm{sl}(2,\mathbf{C})$, Invent. Math. 105 (1991) 473

[19] E S Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005) 554

[20] G Naot, The universal Khovanov link homology theory, Algebr. Geom. Topol. 6 (2006) 1863

[21] D Roseman, Reidemeister-type moves for surfaces in four-dimensional space, from: "Knot theory (Warsaw, 1995)" (editors V F R Jones, P T Joanna Kania-Bartoszyńska Jozef H. Przytycki, V G Turaev), Banach Center Publ. 42, Polish Acad. Sci. (1998) 347

[22] B Webster, Khovanov–Rozansky homology via a canopolis formalism, Algebr. Geom. Topol. 7 (2007) 673