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A Khovanov stable homotopy type for colored links

Andrew Lobb, Patrick Orson and Dirk Schütz

Algebraic & Geometric Topology 17 (2017) 1261–1281

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.

Khovanov, flow category, stable homotopy type
Mathematical Subject Classification 2010
Primary: 57M27
Received: 27 April 2016
Revised: 12 August 2016
Accepted: 21 August 2016
Published: 14 March 2017
Andrew Lobb
Department of Mathematical Sciences
Durham University
Lower Mountjoy
Stockton Road
United Kingdom
Patrick Orson
Département de Mathématiques
Université du Québec à Montréal
Montréal, Québec
H3C 3P8
Dirk Schütz
Department of Mathematical Sciences
Durham University
Lower Mountjoy
Stockton Road
United Kingdom