| The core problem in Knot Theory is how to distinguish whether two links are equivalent or not.Knot invariants are important tools.The certain invariants are homology,genus,unknotting number,linking number and so on.Compared with the Jones polynomial,the Khovanov homology of a link is a stronger invariant.At present,the methods to calculate Khovanov homology include the definition method,the long exact sequence,the spectral sequence and so on.The methods commonly used in literature are the exact sequence and the spectral sequence.If we calculate directly from the definition,as the number of crossings increases,it quickly becomes difficult to calculate.Although it is relatively easy to calculate the Khovanov homology of links with fewer crossings such as Hopf links.However,it is still quite difficult to calculate the Khovanov homology of any infinite families of links.This paper mainly studies the Khovanov homology of certain infinite families of torus links with different coefficients.The main results are as follows:The Khovanov homology for each of an infinite family of T（3,q）over Z2is obtained for q≥3.The results can be divided into three cases q=3N,q=3N+1,and q=3N+2.Then the Khovanov homology over Q for each of the infinite family of links T（4,q）is obtained for q≥4.It is found that its homology depends on q.We discuss the cases q=4N,q=4N+1,q=4（N+1）-2 and q=4（N+1）-1respectively.Firstly,c is obtained,and the corresponding spectral sequence of the first page is written out.Then,the existence of generators is determined by using the properties of linking numbers and Lee homology,and the homology of the remaining positions is determined by using the exact sequence of i and j.Then we turn to the more complex infinite family of links T（5,q）.We propose a conjecture. |
