2018/12/26 by Taketo Sano, Sano, Taketo
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.1812.10258
Lee homology (a variant of Khovanov homology) over ℚ possesses the "canonical generators" as its basis. The generators (Lee's classes) [α(D, o)] are constructed combinatorially from an oriented link diagram D, one for each alternative orientation o on D. Let R be an integral domain. There exists a family of link homology theory \ Hc(-; R) \c ∈ R, where Khovanov's theory corresponds to c = 0 and Lee's theory corresponds to c = 2. For each c ∈ R ∖ 0, Lee's classes [α(D, o)] can be defined as elements in Hc(D; R), but when c is not invertible then they do not form a basis; in fact they are divisible by c-powers. We define the c-divisibility kc(D) of [α(D, o)] with o the given orientation of D. For any link L and its diagram D, we prove that sc(L) := 2kc(D) + w(D) - r(D) + 1 is a link invariant, where w is the writhe, and r is the number of Seifert circles. We pose the question whether sc coincides with Rasmussen's s-invariant. There are several evidences that support the affirmative answer. For instance, sc is a link concordance invariant, and the Milnor conjecture can be reproved using sc. Also for the special case (R, c) = (ℚ[h], h), our sc actually coincides with s as knot invariants.