Khovanov Homology Of Some Virtual Links

The essential problem in Knot Theory is how to distinguish distinct knots or links.Knot invariants are an important tool for measuring whether two knots or links are equivalent.At present,there are many knot invariants,such as homology,knot polynomials,the unknotting number,the bridge number and the genus.The Khovanov homology is the categorization of Jones polynomials and an important bridge between graph and algebra.The methods of determining the Khovanov homology of a knot or link are mainly definition,matrix,exact sequence,spectral sequence and so on.It’s easy to determine some knots or links with fewer crossings,such as the Hopf link,the trefoil,the figure-eight knot.However,the calculation process will become complicated with the number of crossings increasing.Now,there are relatively few research results on knots or links with more than 11 crossings.The virtual links can be regarded as the promotion of the classic link in the thickened surface.So far,there is no research result on the Khovanov homology of the infinite family of virtual links.In this paper,three new infinite families of virtual links are studied,and the Khovanov homology with Z₂ coefficient are determined.The main results are as follows:The paper first propose and prove the Khovanov homology for a kind of infinite family of virtual link Kn with Z₂ coefficient.Kn has a virtual crossing and n classic crossings.Then we increase the number of virtual crossings and compute the Khovanov homology for the infinite family of virtual links Kn with Z₂ coefficient.Kn has 2 virtual crossings and n classic crossings.Finally,we increase 2 crossings of the（3,q）torus links to virtual crossings to obtain a new type of infinite family of virtual links T（3,q）.We find its Khovanov homology is related to q.So we divide into 3 families for computing according to q=3N+1,q=3N+2,q=3（N+1）.In the process of calculation,it is find that the subcomplex of T（3,q）is also a kind of new infinite family virtual links.We denote it as T（3,q）.The Khovanov homology is also related to q.we also divide into 3 families for computing.We compute the Khovanov homology for some T（3,q）and some T（3,q）with Z₂ coefficient and give the conjecture of Khovanov homology for T（3,q）and T（3,q）with Z₂ coefficient.