A Khovanov stable homotopy type for colored links

Lobb, Andrew¹ ; Orson, Patrick² ; Schütz, Dirk¹

¹Department of Mathematical Sciences, Durham University, Lower Mountjoy, Stockton Road, Durham, DH1 3LE, United Kingdom
²Département de Mathématiques, Université du Québec à Montréal, Montréal, Québec, H3C 3P8, Canada

Algebraic and Geometric Topology, Tome 17 (2017) no. 2, pp. 1261-1281

Résumé

We extend Lipshitz and Sarkar’s definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. Given an assignment c (called a coloring) of a positive integer to each component of a link L, we define a stable homotopy type Xcol(Lc) whose cohomology recovers the c–colored Khovanov cohomology of L. This goes via Rozansky’s definition of a categorified Jones–Wenzl projector Pn as an infinite torus braid on n strands.

We then observe that Cooper and Krushkal’s explicit definition of P2 also gives rise to stable homotopy types of colored links (using the restricted palette {1,2}), and we show that these coincide with Xcol. We use this equivalence to compute the stable homotopy type of the (2,1)–colored Hopf link and the 2–colored trefoil. Finally, we discuss the Cooper–Krushkal projector P3 and make a conjecture of Xcol(U3) for U the unknot.

Détail

DOI : 10.2140/agt.2017.17.1261

Classification : 57M27
Keywords: Khovanov, flow category, stable homotopy type

Bibliographie

[1] D Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443 | DOI

[2] H J Baues, M Hennes, The homotopy classification of (n−1)–connected (n+3)–dimensional polyhedra, n ≥ 4, Topology 30 (1991) 373 | DOI

[3] B Cooper, V Krushkal, Categorification of the Jones–Wenzl projectors, Quantum Topol. 3 (2012) 139 | DOI

[4] D Jones, A Lobb, D Schütz, An sln stable homotopy type for matched diagrams, preprint (2015)

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

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

[7] T Lawson, R Lipshitz, S Sarkar, Khovanov homotopy type, Burnside category, and products, preprint (2015)

[8] R Lipshitz, S Sarkar, A Khovanov stable homotopy type, J. Amer. Math. Soc. 27 (2014) 983 | DOI

[9] R Lipshitz, S Sarkar, A Steenrod square on Khovanov homology, J. Topol. 7 (2014) 817 | DOI

[10] L Rozansky, An infinite torus braid yields a categorified Jones–Wenzl projector, Fund. Math. 225 (2014) 305 | DOI

[11] D Schütz, KnotJob, software (2015)

[12] M Willis, Stabilization of the Khovanov homotopy type of torus links, preprint (2015)

[13] M Willis, A colored Khovanov homotopy type for links, and its tail for the unknot, preprint (2016)