Research On Three Types Of Invariants Of Virtual Knots And Links

Virtual knot theory is a generalization of classical knot that Kauffman discovered in 1996.Classical knot theory is the study of problems related to the embedding of cir-cles into three-dimensional space,and virtual knot theory is the study of problems re-lated to the embedding of circles into a thickened surface Sg×R(Sg is a closed surface of genus g,R is number line).Zhiqing Yang and Jifu Xiao discussed three invariants of classical knot that are the unoriented link invariant,oriented link invariant,regional in-variant in their papers.By giving the corresponding definition of the resolving patterns at the virtual crossing and the invariant around the region of the virtual crossing,this paper generalizes the three invariants to virtual knot.Results are as follows:The resolving patterns of classical crossings and the invariant relation of the re-gion around classical crossings in the projection of the virtual link are as the same as their results.For unoriented virtual link,I define the ways to resolve the virtual crossing that are W+d?+d?=0(the same component),and W+d'?+d'?=0(different component)in this paper.Later,we get the properties and theorems that the unoriented virtual invariant satisfies.For oriented virtual link,I define the ways to resolve the virtual crossing that are Ev+C12E+C22W+C32HC+C42HT+d12VC+d22VT=0(one component),and Ev+C'12E+c'22W+d'12S+d'22N=0(different component)in this paper.Based on this,we obtain the properties and theorems that the oriented invariant of the virtual link satisfies.For the regional invariant of virtual link,I define the relationship around a virtual crossing that is r':ax'+by'+cz'+dw'=0 and prove that LT(D)is a virtual regional invariant when the relationship {xz=yw,x'=z',z'z'=w'y',x/w+z/w-yx'/wy'=y'/z',1-zw'/wz'=xw'/wz'-yw'/wy'} is satisfied in this paper.