标题： 关于链同调和扭结曲面的讲义 显示英文标题

标题： Lecture notes on link homologies and knotted surfaces

摘要： 链同调理论（如纽结弗洛尔同调和柯霍诺夫同调）已成为研究纽结和链不可或缺的工具，包括强大的四维障碍。 这些笔记基于2024年佐治亚拓扑暑期学校讲授的课程，讨论这些工具包通过分配给链 cobordisms 的同态，关于四维空间中的曲面本身所说的内容。 我们首先简要概述这些理论（重点在于它们的共同形式属性），并调查它们在扭结曲面方面的应用。 随后，我们介绍柯霍诺夫同调（着眼于其 cobordism 映射），讨论柯霍诺夫和巴-纳坦同调的实用计算技术，并概述巴-纳坦范畴在此故事中的作用。 显示英文摘要

摘要： Link homology theories (such as knot Floer homology and Khovanov homology) have become indispensable tools for studying knots and links, including powerful 4-dimensional obstructions. These notes, based on lectures given at the 2024 Georgia Topology Summer School, discuss what these toolkits say about surfaces in 4-space themselves, via the homomorphisms assigned to link cobordisms. We begin with a brief overview of these theories (focusing on their shared formal properties) and survey some of their applications to knotted surfaces. Afterwards, we give an introduction to Khovanov homology (with an eye towards its cobordism maps), discuss hands-on computational techniques for Khovanov and Bar-Natan homology, and outline the role of the Bar-Natan category in this story.