Why you can't tie knots in four dimensions
Phys.org
February 26, 2026
AI-Generated Deep Dive Summary
Tying knots in four dimensions may seem like an abstract concept, but it has profound implications for understanding the geometry of higher-dimensional spaces. While we are familiar with three-dimensional space, the idea of a fourth dimension often refers to time or spacetime, as introduced by Einstein's theory of relativity. This article explores how knot theory—a mathematical field that studies knots and their properties—takes on new dimensions when extended into four-dimensional space.
In three dimensions, knots can be tangled in complex ways, but they cannot always be untangled without cutting the string. However, in four dimensions, the extra spatial dimension allows for more flexibility. Knots in four dimensions can be manipulated and transformed in ways that are impossible in our everyday three-dimensional world. This property is not just a mathematical curiosity; it has real-world applications in fields like quantum computing, where understanding higher-dimensional structures could help design new materials or solve computational problems.
The concept of four dimensions challenges our intuitive understanding of space and geometry. Unlike the fixed grid of three-dimensional space, four-dimensional space allows for greater freedom and movement. For example, a knot that appears entangled in three dimensions can be "untied" by moving it through the fourth dimension. This ability to untie knots without cutting the string highlights the unique properties of higher-dimensional spaces and opens up new possibilities for mathematical exploration.
Understanding four-dimensional geometry is not just an academic exercise; it has practical implications for physics, computer science, and other disciplines. By studying how objects behave in higher dimensions, scientists can gain insights into complex systems that are difficult to model in three dimensions alone. For instance, the principles of knot theory in four dimensions may help explain the behavior
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Originally published on Phys.org on 2/26/2026