A Khovanov stable homotopy type for colored links

Lobb, Andrew; Orson, Patrick; Schuetz, Dirk

Abstract

We extend Lipshitz-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 positive integer to each component of a link L, we define a stable homotopy type X_col(L_c) whose cohomology recovers the c-colored Khovanov cohomology of L. This goes via Rozansky's definition of a categorified Jones-Wenzl projector P_n as an infinite torus braid on n strands. We then observe that Cooper-Krushkal's explicit definition of P_2 also gives rise to stable homotopy types of colored links (using the restricted palette {1, 2}), and we show that these coincide with X_col. 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 P_3 and make a conjecture of X_col(U_3) for U the unknot.

Citation

Lobb, A., Orson, P., & Schuetz, D. (2017). A Khovanov stable homotopy type for colored links. Algebraic & geometric topology, 17(2), 1261-1281. https://doi.org/10.2140/agt.2017.17.1261

• Journal Article Type: Article
• Acceptance Date: Aug 21, 2016
• Online Publication Date: Mar 14, 2017
• Publication Date: Mar 14, 2017
• Deposit Date: Aug 22, 2016
• Publicly Available Date: Apr 21, 2017
• Journal: Algebraic and Geometric Topology
• Print ISSN: 1472-2747
• Electronic ISSN: 1472-2739
• Publisher: Mathematical Sciences Publishers (MSP)
• Peer Reviewed: Peer Reviewed
• Volume: 17
• Issue: 2
• Pages: 1261-1281
• DOI: https://doi.org/10.2140/agt.2017.17.1261
• Public URL: https://durham-repository.worktribe.com/output/1377526