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A Khovanov stable homotopy type for colored links.

Lobb, Andrew and Orson, Patrick and Schuetz, Dirk (2017) 'A Khovanov stable homotopy type for colored links.', Algebraic and geometric topology., 17 (2). pp. 1261-1281.


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.

Item Type:Article
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Publisher statement:First published in Algebraic & Geometric Topology in 17 (2017) 1261–1281, published by Mathematical Sciences Publishers. © 2017 Mathematical Sciences Publishers. All rights reserved.
Date accepted:21 August 2016
Date deposited:22 August 2016
Date of first online publication:14 March 2017
Date first made open access:No date available

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