A Glimpse of the Khovanov Homology of T(2,n) Via Long Exact Sequence

Abstract

Khovanov homology is a powerful link invariant: a categorification of the Jones polynomial that enjoys a rich and beautiful algebraic structure. This homology theory has been extensively studied and it has become an ubiquitous topic in contemporary knot theory research. In the same spirit, the Kauffman skein relation, which allows to define the Kauffman bracket polynomial up to normalization of the unknot, can be categorified by means of a long exact sequence. In an expository style, in this article we present how to build Khovanov homology from the Kauffman bracket polynomial and construct its long exact sequence. Furthermore, we present a deviceful and practical way in which this long exact sequence can be used for the computation of the Khovanov homology of torus links of the type $T(2,n)$. This article serves as a partial translation of a Spanish paper to be published on occasion of the Encuentro Internacional de Matem\'aticas (International Meeting of Mathematics) celebrated at the Universidad del Atl\'antico in Barranquilla, Colombia in November 2023. This paper offers a first look into the world of Khovanov homology by constructing it from the Kauffman bracket polynomial, as it was first done by Oleg Viro. Moreover, it gives the reader references for further studies from leading experts such as D. Bar-Natan, M. Khovanov, S. Mukherjee, J. Przytycki, and A. Shumakovitch, among others. In particular, one of the main objectives in publishing this article (and this partial translation) is to popularize research in knot theory, more specifically on Khovanov homology in Colombia, and Latin-America in general, acting as a language bridge given that most of the literature is in English.