**1. Introduction**

Khovanov homology was introduced by Mikhail Khovanov in 2000 in Reference [1] as a categorification of the Jones polynomial, which was introduced by Jones in [2]. His construction, using geometrical and topological objects instead of polynomials, was so interesting that it offered a completely new approach to tackle problems in low-dimensional topology.

Khovanov homology plays a vital role in developing several important results in the field of knot theory. Soon after the discovery of Khovanov homology, Bar-Natan proved in Reference [3] that Khovanov's invariant is stronger than the Jones polynomial. He also proved that the graded Euler characteristic of the chain complex of a link *L* is the un-normalized Jones polynomial of that link. In 2005, Bar-Natan extended the Khovanov homology of links to tangles, cobordisms, and two-knots [4]. In [5] Bar-Natan gave a fast way of computing the Khovanov homology. In 2013, Ozsvath, Rasmussen, and Szabo introduced the odd Khovanov homology by using exterior algebra instead of symmetric algebra [6]. Gorsky, Oblomkov, and Rasmussen gave some results on stable Khovanov homology of torus links in Reference [7]. Putyra introduced a triply graded Khovanov homology and used it to prove that odd Khovanov homology is multiplicative with respect to disjoint unions and connected sums of links Reference [8]. Manion gave rational Khovanov homology of three-strand pretzel links in 2011 [9]. Nizami, Mobeen, and Ammara gave Khovanov homology of some families of braid links in Reference [10]. Nizami, Mobeen, Sohail, and Usman gave Khovanov homology and graded Euler characteristic of 2-strand braid links in [11].

In Reference [12], Marko used a long exact sequence to prove that the Khovanov homology groups of the torus link *<sup>T</sup>*(*n*; *m*) stabilize as *m* → ∞. A generalization of this result to the context of tangles came in the form of Reference [13], where Lev Rozansky showed that the Khovanov chain complexes for torus braids also stabilize (up to chain homotopy) in a suitable sense to categorify the Jones–Wenzl projectors. At roughly the same time, Benjamin Cooper and Slava Krushkal gave an alternative construction for the categorified projectors in Reference [14]. These results, along with connections between Khovanov homology, HOMFLYPT homology, Khovanov–Rozansky homology, and the representation theory of rational Cherednik algebra (see [15]) have led to conjectures about the structure of stable Khovanov homology groups in limit *Kh*(*T*(*n*; 1)) (see [15], and results along these lines in Reference [16]). More recently, in Reference [17], Robert Lipshitz and Sucharit Sarkar introduced the Khovanov homotopy type of a link *L*. This is a link invariant taking the form of a spectrum whose reduced cohomology is the Khovanov homology of *L*.

Although computing the Khovanov homology of links is common in the literature, no general formulae have been given for all families of knots and links. In this paper, we give Khovanov homology of the three-strand braid links Δ2*k*+1, <sup>Δ</sup>2*k*+1*x*2, and <sup>Δ</sup>2*k*+1*x*1, where Δ is the Garside element *x*1*x*2*x*1. Particularly, we focus on the top homology groups.
