Khovanov homology via 1-tangle diagrams in the annulus

David Boozer

Published 2021-02-22Version 1

We show that the reduced Khovanov homology of a link $L$ in $S^3$ can be expressed as the homology of a chain complex constructed from a description of $L$ as the closure of a 1-tangle diagram in the annulus. The chain complex is constructed using a resolution of the 1-tangle diagram into planar tangles in a manner analogous to ordinary reduced Khovanov homology, but with some novel features. In particular, unlike for ordinary Khovanov homology, terms appear in the differential that are products of linear maps corresponding to pairs of saddles obtained from the resolution. We also use the chain complex to construct a spectral sequence that converges to reduced Khovanov homology.