TL;DR
This paper introduces a new invariant for three-component link homotopy in four-dimensional space, extending previous work on two-component links, and demonstrates its effectiveness with specific examples.
Contribution
The paper constructs a novel three-component link homotopy invariant and develops tools to distinguish complex link maps, also discussing potential generalizations to n-component links.
Findings
Constructed two link maps with identical images but different homotopy classes.
Developed a new invariant capable of distinguishing non-homotopic three-component links.
Extended the invariant to three-component annular link maps.
Abstract
In 2019, Schneidermann and Teicher showed that the Kirk invariant classifies two-component link maps of two-spheres in the four-sphere up to link homotopy. In this paper, we construct a three-component link homotopy invariant. We construct two link maps where each component has the same image, and apply our invariant to prove that nevertheless they are not link homotopic. We develop tools to help distinguish between three-component link maps. We then construct a similar invariant for three-component annular link maps. Towards the end of the paper we discuss how to generalise to an -component link map invariant.
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Taxonomy
TopicsGeometric and Algebraic Topology · Topological and Geometric Data Analysis · Homotopy and Cohomology in Algebraic Topology