A Glimpse of the Khovanov Homology of T(2,n) Via Long Exact Sequence
A Glimpse of the Khovanov Homology of T(2,n) Via Long Exact Sequence
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通过长精确序列瞥见 T(2,n) 的霍瓦诺夫同调
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发表时间：
2023
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通讯作者：
Gabriel Montoya
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作者：
Gabriel Montoya
霍瓦诺夫同调是一个强大的链接不变量：琼斯多项式的分类，具有丰富而美丽的代数结构。这种同源理论得到了广泛的研究，成为当代纽结理论研究中普遍存在的课题。本着同样的精神，考夫曼绞线关系允许定义考夫曼括号多项式直至结的标准化，可以通过长精确序列进行分类。在本文中，我们以说明性的方式介绍了如何从考夫曼括号多项式构建霍瓦诺夫同调并构造其长精确序列。此外，我们提出了一种巧妙且实用的方法，可以使用这个长的精确序列来计算 $T(2,n)$ 类型的环面链接的 Khovanov 同源性。本文是西班牙论文的部分翻译，该论文将于 2023 年 11 月在哥伦比亚巴兰基亚的大西洋大学庆祝 Encuentro Internacional de Matem'aticas（国际数学会议）之际发表。本文通过从考夫曼括号多项式构造 Khovanov 同调性，首次介绍了 Khovanov 同调的世界，正如 Oleg Viro 首次完成的那样。此外，它还为读者提供了 D. Bar-Natan、M. Khovanov、S. Mukherjee、J. Przytycki 和 A. Shumakovitch 等领先专家进一步研究的参考资料。特别是，发表这篇文章（以及部分翻译）的主要目标之一是普及纽结理论的研究，更具体地说是关于哥伦比亚和整个拉丁美洲的霍瓦诺夫同调性的研究，鉴于大多数文献都是英文的，充当语言桥梁。
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) to be 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.