Plumbing is a natural operation in Khovanov homology [article]

Thomas Kindred
2017 arXiv   pre-print
Given a connect sum of link diagrams, there is an isomorphism which decomposes unnormalized Khovanov chain groups for the product in terms of normalized chain groups for the factors; this isomorphism is straightforward to see on the level of chains. Similarly, any plumbing x*y of Kauffman states carries an isomorphism of the chain subgroups generated by the enhancements of x*y, x, y: C_R(x*y)→(C_R,p→1(x)⊗C_R,p→1(y))⊕(C_R,p→0(x)⊗C_R,p→0(y)). We apply this plumbing of chains to to prove that
more » ... homogeneously adequate state has enhancements X^± in distinct j-gradings whose A-traces (which we define) represent nonzero Khovanov homology classes over F_2, and that this is also true over Z when all A-blocks' state surfaces are two-sided. We construct X^± explicitly.
arXiv:1705.01931v1 fatcat:hnrohbcg7neo7h7orhdhethnvy