Abstract
Khovanov homology is a categorication of the Jones polynomial. It consists of graded chain complexes which, up to chain homotopy, are link invariants, and whose graded Euler characteristic is equal to the Jones polynomial of the link. In this article we give some Khovanov homology groups of 3-strand braid links Δ 2 k + 1 = x 1 2 k + 2 x 2 x 1 2 x 2 2 x 1 2 ⋯ x 2 2 x 1 2 x 1 2 , Δ 2 k + 1 x 2 , and Δ 2 k + 1 x 1 , where Δ is the Garside element x 1 x 2 x 1 , and which are three out of all six classes of the general braid x 1 x 2 x 1 x 2 ⋯ with n factors.
Highlights
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]
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]
Summary
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]. Mobeen, Sohail, and Usman gave Khovanov homology and graded Euler characteristic of 2-strand braid links in [11]. 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]). 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.
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