Generalization of quandles and their applications

作为光团的推广，我们研究了双光团和对称光团。用一套公理重新定义了双轴的概念，并引入了另外两种结构：t型结构和v型结构。这些结构与虚结和扭曲虚结有关。我们还给出了一种从给定的堆中构造双堆的方法。通过这种方法，可以方便有效地构造多个双角。研究了一种对称纠缠的表示形式，给出了一种获得流形结基本对称纠缠表示形式的方法。

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.