- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:39
Quantum Topology
Quantum Topol.
QT
1663-487X
1664-073X
General
10.4171/QT
http://www.ems-ph.org/doi/10.4171/QT
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
6
2015
4
Equivariant Khovanov–Rozansky homology and Lee–Gornik spectral sequence
Hao
Wu
George Washington University, WASHINGTON, UNITED STATES
Equivariant Khovanov–Rozansky homology, homological thickness, Lee– Gornik spectral sequence, exact couple
Lobb observed in [8] that each equivariant $\mathfrak {sl} (N)$ Khovanov–Rozansky homology over $\mathbb C [a]$ admits a standard decomposition of a simple form. In the present paper, we derive a formula for the corresponding Lee–Gornik spectral sequence in terms of this decomposition. Based on this formula, we give a simple alternative definition of the Lee–Gornik spectral sequence using exact couples. We also demonstrate that an equivariant $\mathfrak {sl} (N)$ Khovanov–Rozansky homology over $\mathbb C [a]$ can be recovered from the corresponding Lee–Gornik spectral sequence via this formula. Therefore, these two algebraic invariants are equivalent and contain the same information about the link. As a byproduct of the exact couple construction, we generalize Lee's endomorphism on the rational Khovanov homology to a natural $\bigwedge^\ast \mathbb C^{N-1}$-action on the $\mathfrak sl (N)$ Khovanov–Rozansky homology. A numerical link invariant called torsion width comes up naturally in our work. It determines when the corresponding Lee–Gornik spectral sequence collapses and is bounded from above by the homological thickness of the $\mathfrak {sl} (N)$ Khovanov–Rozansky homology. We use the torsion width to explain why the Lee spectral sequences of certain H-thick links collapse so fast.
Manifolds and cell complexes
General
515
607
10.4171/QT/70
http://www.ems-ph.org/doi/10.4171/QT/70