Plumbing essential states in Khovanov homology
||knot, link, state, spanning surface, essential, alternating, checkerboard, plumbing, Murasugi sum, Khovanov homology, adequate, homogeneous|
We prove that every homogeneously adequate Kauffman state has enhancements
X± in distinct j-gradings whose traces (which we define) represent nonzero Khovanov homology classes over Z/2Z; this is also true over Z when all A-blocks' state surfaces are two-sided. A direct proof constructs
X± explicitly. An alternate proof, reflecting the theorem's geometric motivation, applies a plumbing (Murasugi sum) operation that has been adapted to the context of Khovanov homology.
Thank you to the referee for suggesting many improvements to this paper's details and overall structure
Department of Mathematics, University of Nebraska-Lincoln, Lincoln,
NE 68588, USA