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Evaluations of annular Khovanov--Rozansky homology

Eugene Gorsky, Paul Wedrich

Abstract

We describe the universal target of annular Khovanov-Rozansky link homology functors as the homotopy category of a free symmetric monoidal category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use to characterize the annular invariants of Coxeter braids. Further, we prove the existence of symmetric group actions on the Khovanov-Rozansky invariants of cabled tangles and we introduce spectral sequences that aid in computing the homologies of generalized Hopf links. Finally, we conjecture a characterization of the horizontal traces of Rouquier complexes of Coxeter braids in other types.

Evaluations of annular Khovanov--Rozansky homology

Abstract

We describe the universal target of annular Khovanov-Rozansky link homology functors as the homotopy category of a free symmetric monoidal category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use to characterize the annular invariants of Coxeter braids. Further, we prove the existence of symmetric group actions on the Khovanov-Rozansky invariants of cabled tangles and we introduce spectral sequences that aid in computing the homologies of generalized Hopf links. Finally, we conjecture a characterization of the horizontal traces of Rouquier complexes of Coxeter braids in other types.

Paper Structure

This paper contains 45 sections, 89 theorems, 157 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The annular category
  3. Spectral sequences
  4. Positive Coxeter braids
  5. Other Coxeter lifts
  6. Organization of the paper
  7. The classical story
  8. The skein of the annulus
  9. Universal Hecke trace and symmetric functions
  10. Coxeter braids
  11. From skein to the center of Hecke algebra
  12. Generalized Hopf links
  13. General facts about symmetric monoidal categories
  14. A free symmetric monoidal category
  15. Schur functors and evaluation
  16. ...and 30 more sections

Key Result

Theorem 1.1

The Karoubi completion (or bounded homotopy category) of the category of positive annular webs and foams is equivalent to (the bounded homotopy category of) the free symmetric monoidal graded Karoubian category $\hat{\hbox{\bfseries {\upshape {P}}}}$ generated by a single object $E$ (corresponding t

Figures (5)

  • Figure 1: A standard Young tableau and the content filling for the Young diagram for $\lambda=(4,3,3,2)$.
  • Figure 2:
  • Figure 3: The result of substituting a single column $(C^{i,*}, d_v)$ in a bicomplex by a homotopy equivalent complex $(D^{i,*},d)$ along chain homotopy equivalences $f$ and $g$ with $g \circ f + d_v \circ h + h \circ d_v =0$.
  • Figure 4:
  • Figure :

Theorems & Definitions (207)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Definition 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 197 more