## Algebraic & Geometric Topology

### Functoriality for the $\mathfrak{su}_3$ Khovanov homology

David Clark

#### Abstract

We prove that the categorified $su3$ quantum link invariant is functorial with respect to tangle cobordisms. This is in contrast to the categorified $su2$ theory, which was not functorial as originally defined.

We use methods of Morrison and Nieh and Bar-Natan to construct explicit chain maps for each variation of the third Reidemeister move. Then, to show functoriality, we modify arguments used by Clark, Morrison and Walker to show that induced chain maps are invariant, up to homotopy, under Carter and Saito’s movie moves.

#### Article information

Source
Algebr. Geom. Topol., Volume 9, Number 2 (2009), 625-690.

Dates
Revised: 2 March 2009
Accepted: 5 March 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796995

Digital Object Identifier
doi:10.2140/agt.2009.9.625

Mathematical Reviews number (MathSciNet)
MR2482322

Zentralblatt MATH identifier
1165.57005

#### Citation

Clark, David. Functoriality for the $\mathfrak{su}_3$ Khovanov homology. Algebr. Geom. Topol. 9 (2009), no. 2, 625--690. doi:10.2140/agt.2009.9.625. https://projecteuclid.org/euclid.agt/1513796995

#### References

• D Bar-Natan, Khovanov's homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443–1499
• D Bar-Natan, Fast Khovanov homology computations, J. Knot Theory Ramifications 16 (2007) 243–255
• J S Carter, J H Rieger, M Saito, A combinatorial description of knotted surfaces and their isotopies, Adv. Math. 127 (1997) 1–51
• J S Carter, M Saito, Reidemeister moves for surface isotopies and their interpretation as moves to movies, J. Knot Theory Ramifications 2 (1993) 251–284
• D Clark, S Morrison, K Walker, Fixing the functoriality of Khovanov homology, Geom. Topol. 13 (2009) 1499–1582
• S I Gelfand, Y I Manin, Methods of homological algebra, Springer, Berlin (1996) Translated from the 1988 Russian original
• M Jacobsson, An invariant of link cobordisms from Khovanov homology, Algebr. Geom. Topol. 4 (2004) 1211–1251
• M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359–426
• M Khovanov, sl(3) link homology, Algebr. Geom. Topol. 4 (2004) 1045–1081
• G Kuperberg, Spiders for rank $2$ Lie algebras, Comm. Math. Phys. 180 (1996) 109–151
• M Mackaay, P Vaz, The universal ${\rm sl}\sb 3$–link homology, Algebr. Geom. Topol. 7 (2007) 1135–1169
• S Morrison, A Nieh, On Khovanov's cobordism theory for $\mathfrak{su}\sb 3$ knot homology, J. Knot Theory Ramifications 17 (2008) 1121–1173
• D Roseman, Reidemeister-type moves for surfaces in four-dimensional space, from: “Knot theory (Warsaw, 1995)”, (V F R Jones, J Kania-Bartoszyńska, J H Przytycki, P Traczyk, V G Turaev, editors), Banach Center Publ. 42, Polish Acad. Sci., Warsaw (1998) 347–380
• B Webster, Khovanov–Rozansky homology via a canopolis formalism, Algebr. Geom. Topol. 7 (2007) 673–699
• Wiktionary, the free online dictionary (2008), sv “metropolis” Available at \setbox0\makeatletter\@url http://en.wiktionary.org/wiki/metropolis {\unhbox0