Categorification of quantum invariants of 3-manifolds

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This study gave a fundamental theory contributing to attack the famous open problem: "Categorificaiton of quantum invariants of 3-manifolds". The key point to solve the problem is that we consider the possibility to define invariants under two kinds of operations, denoted by K1 and K2. The main three results are as follows. (1) We reveal that K1 corresponds to sliding grades of a known categorification. (2) We obtain the objects that are invariant under K2 by redefining a generalization of known categorification. (3) We gave an explicit description of a geometric operation derived from (2).