The k-colored sl(N) homology of T(2,2m+1) stabilizes as m to infinity to the integral homology of the free loop space of Gr(k,N).
The minimal Rickard complexes of braids on two strands
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abstract
The Rickard complex of a braid with strands colored by positive integers is a chain complex of singular Soergel bimodules. The complex determines the colored triply-graded homology and colored sl(N) homology of the braid closure, when closure is color-compatible. For each braid on two strands with any colors, we construct a minimal complex that is homotopy equivalent to its Rickard complex. It is not obtained by laborious simplification; instead, it is defined directly by explicit formulas obtained by educated guesswork and reverse engineering.
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2026 1verdicts
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Link homology and loop homology
The k-colored sl(N) homology of T(2,2m+1) stabilizes as m to infinity to the integral homology of the free loop space of Gr(k,N).