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Why is knot theory considered part of algebraic topology?

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topology Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.

Knot theory is considered a branch of algebraic topology because it is fundamentally involved in the study of three-dimensional manifolds. It examines how knotted copies of a circle can be embedded within three-dimensional space.1

The core objective of knot theory is to determine whether a closed curve is truly knotted or can be continuously deformed into a standard unknotted circle, and whether two given curves represent the same knot. By analyzing these topological properties, mathematicians classify knots based on their complexity, such as the minimum number of crossings in a projection.2

While knot theory originated as a study of closed curves, its classification and analysis have become deeply intertwined with algebraic methods, particularly following the introduction of invariants like the Jones polynomial. These tools have further connected the field to quantum field theory and hyperbolic geometry, demonstrating the power of algebraic approaches in understanding the structure of space.2