组合纽结论是通过平面投影图来研究纽结和链环的理论。研究图和链环之间的联系是一个有趣的课题，最常见的联系是图的图多项式与链环的链环多项式之间的联系。比如，交错链环的Jones多项式（由V. F. R. Jones提出，Jones由于该项工作在1990年国际数学家大会上荣获Fields奖。）是Tutte多项式的特殊情形。本项目主要研究非交错链环和棋盘可着色虚拟链环Jones多项式的系数，并探索虚拟链环的Jones多项式以及更广的一些多项式（比如，广义Yamada多项式）的系数、次数和跨度以及相关问题。这些纽结多项式与图论中的图的Tutte多项式以及带子图的Bollobás-Riordan多项式有密切关联。拟从图和图多项式入手，综合运用图论和组合纽结论中的方法和技巧，期望得到Jones多项式系数的几何解释。

The combinatorial knot theory is the theory of studying knots and links by using diagrams in plane. It is an interesting subject to study the relationship between graphs and links. The relationship between graph polynomials of graphs and link polynomials of links is the most common. For example, the Jones polynomial (which was proposed by V. F. R. Jones. Jones won the Fields Award at the 1990 International Conference of Mathematicians for this work.) of an alternating link is a specialization of the Tutte polynomial. In this project, we will mainly focus on the coefficients of Jones polynomials of non-alternating links and checkerboard colorable virtual links. We explore the coefficients, degrees and spans of Jones polynomial and some more general polynomials (such as generalized Yamada polynomial) of virtual links and some related problems. These knot polynomials are closely related to the Tutte polynomial of graphs and the Bollobás-Riordan polynomial of ribbon graphs in graph theory. We intend to start from the graphs and graph polynomials, combine the methods and techniques in graph theory and combinatorial knot theory, and expect to obtain the geometric interpretation of the coefficients of Jones polynomial.