Research on 4-dimensional topology from the viewpoint of graphics and quandle theory

为了描述4-流形的亏格2的Lefschetz纤维的结构，我们引入了一种称为图描述的图解方法。利用这个方法，我们得到了关于亏格2的Lefschetz函数的稳定性的一个定理，它是由D.Auroux得到的一个定理的推广，也是由B.Siebert和G.tin得到的一个定理.对称Quandle是一个具有附加结构的Quandle，称为好对合.介绍了对称双环的同调理论，并利用它定义了4维空间中曲面链环的一个不变量。即使表面连接是不可定向的，也定义了不变量。它可以用来估计曲面连接的最小三元数。

In order to describe a structure of a genus-2 Lefschetz fibration of a 4-manifold, we introduced a graphical method, called the chart description. Using this method, we obtained a theorem on stabilization of genus-2 Lefschetz fibrations, that is a generalization of a theorem due to D. Auroux, and a theorem due to B. Siebert and G. Tian.A symmetric quandle is a quandle with an additional structure, called a good involution. We introduced the homology theory of a symmetric quandle, and using it, we defined an invariant for surface links in 4-space. The invariant is defined even if the surface link is non-orientable. It can be used for estimation of the minimal triple number of a surface-link.