A Khovanov stable homotopy type for colored links
| Autor: | Patrick Orson, Andrew Lobb, Dirk Schütz |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
stable homotopy type
Pure mathematics flow category Khovanov Mathematics::Algebraic Topology 01 natural sciences Mathematics - Geometric Topology Mathematics::K-Theory and Homology Mathematics::Quantum Algebra 0103 physical sciences FOS: Mathematics Braid Algebraic Topology (math.AT) Mathematics - Algebraic Topology 0101 mathematics Unknot Trefoil Mathematics Conjecture Homotopy 010102 general mathematics Geometric Topology (math.GT) Torus Mathematics::Geometric Topology Cohomology 57M27 Hopf link 010307 mathematical physics Geometry and Topology |
| Zdroj: | Algebraic and geometric topology, 2017, Vol.17(2), pp.1261-1281 [Peer Reviewed Journal] Algebr. Geom. Topol. 17, no. 2 (2017), 1261-1281 |
| ISSN: | 1472-2739 1472-2747 |
| DOI: | 10.2140/agt.2017.17.1261 |
| Popis: | 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. Comment: 16 pages, 6 figures |
| Databáze: | OpenAIRE |
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