The Khovanov Homology Of Two Strand （?） And Three Strand Δ_2m Braid Links

Low-dimension topology has attracted considerable attention over the last two decades many new invariants has been introduced in this field like Jones and HOM-FLY polynomials of links and knot. Of these invariants, the most important invariant which has been studied frequently in literature is known as Khovanov homology for knots and links. Mikhail Khovanov [BN03] introduced this invariant in his famous pa-per named "A Categorification of the Jones polynomial." Formally now a days this invariant is called Khovanov Homology. Khovanov assigned to the oriented link dia-gram L a chain complex CT,S (L) the he add a bigraded linear differential of degree (1,0) to this chain complex. The deformation (homotopy)of the link depends only on the iso-topy class of the oriented link diagram L. The bigraded homology group Hr.S(L) of the chain complex Cr,S(D) of an oriented link of plane diagram D, is known as the Kho-vanov Homology. This homology depends on two integer r,s. The integer r denotes the homology degree and integer s denotes the quantum degree. In literature, the study of Khovanov Homology of links is common, but unfortunately there is no general for-mulae available for the Khovanov Homology of Two strand braid links a1m, and Three strand braid link △2m. Here in this thesis we will compute the general formulae for Two and Three strand braid links together a closed form for their graded Euler characteristic. Although Khovanov’s construction is combinatorially from which Khovanov Homol-ogy is algorithmically computed. We shall use a simple way of Bar-Naton;s approch which he introduced in his paper to compute the Khovanov Homomlogy.