Generalization of quandles and their applications

作为quandle的推广，我们研究了双quandle和对称quandle。 双量子的概念被一个公理系统重新定义，并引入了两个额外的结构，称为T-结构和V-结构。这些结构与虚纽结和扭曲虚纽结有关。给出了从一个给定的quandle构造一个双quandle的方法。利用该方法，可以方便有效地构造出多个双量子点。 研究了对称量子点的表示，给出了一种求流形纽结基本对称量子点表示的方法。

As generalizations of a quandle, we studied biquandles and symmetric quandles. The notion of a biquandle was re-defined by a system of axioms, and two additional structures, called T-structure and V-structure, were introduced. These structures are related to virtual knots and twisted virtual knots. We also gave a method of constructing a biquandle from a given quandle. By the method, one can easily and effectively construct many biquandles. Presentation of a symmetric quandle was studied, and a method of obtaining a presentation of the fundamental symmetric quandle of a manifold knot was introduced.