纽结理论中，Fox-染色实际上是一种阐释纽结群（或链环群）到二面体群的表示的初等方法。1999年，Harary与Kauffman引入了纽结与链环的最小色数的概念，引发了大量后续工作。本项目试图在研究中使用申请人及合作者发展的染色集与关联图障碍方法，线性代数方法，与图论技巧相结合，证明或证伪关于行列式不为0的p-可染色链环的最小色数的猜想。进一步地，我们试图回答Nakamura、Nakanishi和Satoh提出的一个涉及p-可染色纽结的交叉指标的不等式的公开问题。

In knot theory, Fox coloring is an elementary method of specifying a representation of a knot group (or a link group) onto the dihedral group. In 1999, Harary and Kauffman introduced the concept "minimum number of colors" which leads to many follow-up works. For this project, we will try to prove or disprove a conjecture on the minimum number of colors of p-colorable links with non-zero determinant. The color set and associated graph method developed by the applicant and his collaborators, and the linear algebraic method will be used in the research, combining with some graph theoretical techniques. Furthermore, we shall attempt to answer an open problem on an inequality involving crossing numbers of p-colorable knots, which raised by Nakamura, Nakanishi and Satoh.