New York Journal of Mathematics
Volume 24 (2018), 588-610


Thomas Kindred

Plumbing essential states in Khovanov homology

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Published: August 27,2018.
Keywords: knot, link, state, spanning surface, essential, alternating, checkerboard, plumbing, Murasugi sum, Khovanov homology, adequate, homogeneous
Subject: 57M27, 57M25

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

Author information

Thomas Kindred:
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA