Study on knot invariants detecting the unknot

本研究主要是关于纽结不变量的结构（可能）检测unknot。更确切地说，我们关注的是特征标簇X（K）的几何结构，它是由与一个纽结K相关联的一些代数多项式的零点定义的。事实上，我们发现由一个特殊方程定义的X（K）的截面（子簇）S（K）可以检测小纽结集合中的解纽结。这一结果可为纽结理论提供一些应用。

This study is concerned with a research on the structure of knot invariants (possibly) detecting the unknot. More precisely, we focus on the geometric structure of the character variety X(K), which is defined by the zeros of some algebraic polynomials associated with a knot K. In fact, we found that the section (subvariety) S(K) of X(K) defined by a special equation can detect the unknot in the set of small knots. This result can give some applications to the knot theory.