STUDY OF TOPOLOGICAL INVARIANTS FOR KNOTS AND THEIR APPLICATIONS

主要有五个结果。我们研究了delta移动如何影响连杆的第一HOMFLY系数多项式。然后我们把它推广到n-move。我们研究了link_gould （LG）多项式，它是一个量子不变量。利用Ishii发现的绞结关系，我们找到了一种方法来构建共享相同LG多项式的结或链接。然后我们用相同的LG多项式给出任意多的2桥结和连接。这些2桥结和链接也共享相同的HOMFLY、Kauffman和2变量Alexander（在链接的情况下）多项式。我们给出了涉及连接数的考夫曼链接多项式的前四个系数多项式和一分量子链接的考夫曼多项式的系数多项式的公式。给出了双编织虚结或虚环的群，并考虑了双编织虚结的某些特殊族的群。已知虚结群的集合与带环结群的集合相同。通过实例讨论了虚结群与带状2-球结群、2-球结群、环结群、3-球结群等结群之间的关系。给出了2根索杆的HOMFLYPT多项式的绞结关系。使用这个，我们已经表明，在第2部分中提到的2桥结或链接的集合也共享相同的2缆HOMFLY多项式。

There are five main results.1.We investigated how a delta move influences the first HOMFLY coefficient polynomials of a link. Then we generalized this to a Cn-move.2.We studied the Links-Gould (LG) polynomial, which is a quantum invariant. Using a skein relation discovered by Ishii, we found a way to construct knots or links sharing the same LG polynomial. Then we gave arbitrarily many 2-bridge knots and links with the same LG polynomial. These 2-bridge knots and links also share the same HOMFLY, Kauffman, and 2-variable Alexander (in case of links) polynomials.3.We give formulas for the first four coefficient polynomials of the Kauffman's link polynomial involving linking numbers and the coefficient polynomials of the Kauffman polynomials of the one- and two-component sublinks.4.Giving a presentation of the group of a 2-braid virtual knot or link, we consider the groups of certain special families of 2-braid virtual knots. It is known that the collection of the virtual knot groups is the same as that of the ribbon torus-knot groups. Using our examples we discuss the relationship among the virtual knot groups and other knot groups such as ribbon 2-sphere-knot groups, 2-sphere-knot groups, torus-knot groups, and 3-sphere-knot groups.5.We give a skein relation for the HOMFLYPT polynomials of 2 cable links. Using this, we have shown that the collection of 2-bridge knots or links mentioned in part 2 also share the same 2-cable HOMFLY polynomial.