A sheaf-theoretic description of Khovanov's knot homology

A sheaf-theoretic description of Khovanov's knot homology Article

Gillam, WD. (2012). A sheaf-theoretic description of Khovanov's knot homology . 21(5), 10.1142/S0218216511010061

We give a description of Khovanov's knot homology theory in the language of sheaves. To do this, we identify two cohomology theories associated to a commutative diagram of abelian groups indexed by elements of the cube {0, 1} n. The first is obtained by taking the cohomology groups of the chain complex constructed by summing along the diagonals of the cube and inserting signs to force d 2 = 0. The second is obtained by regarding the commutative diagram as a sheaf on the cube (in the order-filter topology) and considering sheaf cohomology with supports. Included is a general study of sheaves on finite posets, and a review of some basic properties of knot homology in the language of sheaves. © 2012 World Scientific Publishing Company.