## Algebraic & Geometric Topology

### A type $A$ structure in Khovanov homology

Lawrence Roberts

#### Abstract

Inspired by bordered Floer homology, we describe a type $A$ structure in Khovanov homology, which complements the type $D$ structure previously defined by the author. The type $A$ structure is a differential module over a certain algebra. This can be paired with the type $D$ structure to recover the Khovanov chain complex. The homotopy type of the type $A$ structure is a tangle invariant, and homotopy equivalences of the type $A$ structure result in chain homotopy equivalences on the Khovanov chain complex found from a pairing. We use this to simplify computations and introduce a modular approach to the computation of Khovanov homologies. Several examples are included, showing in particular how this approach computes the correct torsion summands for the Khovanov homology of connect sums. A lengthy appendix is devoted to establishing the theory of these structures over a characteristic-zero ring.

#### Article information

Source
Algebr. Geom. Topol., Volume 16, Number 6 (2016), 3653-3719.

Dates
Accepted: 18 April 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841273

Digital Object Identifier
doi:10.2140/agt.2016.16.3653

Mathematical Reviews number (MathSciNet)
MR3584271

Zentralblatt MATH identifier
1360.57026

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 55N35: Other homology theories

#### Citation

Roberts, Lawrence. A type $A$ structure in Khovanov homology. Algebr. Geom. Topol. 16 (2016), no. 6, 3653--3719. doi:10.2140/agt.2016.16.3653. https://projecteuclid.org/euclid.agt/1510841273

#### References

• M,M Asaeda, J,H Przytycki, A,S Sikora, Categorification of the Kauffman bracket skein module of $I$–bundles over surfaces, Algebr. Geom. Topol. 4 (2004) 1177–1210
• M,M Asaeda, J,H Przytycki, A,S Sikora, Categorification of the skein module of tangles, from “Primes and knots” (T Kohno, M Morishita, editors), Contemp. Math. 416, Amer. Math. Soc. (2006) 1–8
• D Bar-Natan, On Khovanov's categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002) 337–370
• J Green, D Bar-Natan, JavaKh (2005) Available at \setbox0\makeatletter\@url http://katlas.org/wiki/Khovanov_Homology {\unhbox0
• B Keller, Introduction to $A$–infinity algebras and modules, Homology Homotopy Appl. 3 (2001) 1–35
• M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359–426
• M Khovanov, A functor–valued invariant of tangles, Algebr. Geom. Topol. 2 (2002) 665–741
• R Lipshitz, P Ozsvath, D Thurston, Bordered Heegaard Floer homology: Invariance and pairing, preprint (2008)
• L,P Roberts, A type $D$ structure in Khovanov homology, Adv. Math. 293 (2016) 81–145