Categorification of quantum invariants of 3-manifolds

这一研究为攻克著名的公开问题“三维流形的量子不变量范畴”提供了基本的理论基础。解决这个问题的关键是我们考虑了在两种操作下定义不变量的可能性，分别表示为K1和K2。主要的三个结果如下。(1)我们揭示了K1对应于一个已知范畴的滑动等级。(2)通过重新定义已知范畴的推广，得到在K2下不变的对象。(3)给出了(2)中一个几何运算的显式刻画。

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).