Knot complements refer to the three-dimensional space obtained by removing the knot from the three-dimensional sphere. This concept is crucial in knot theory as it allows mathematicians to study the properties and characteristics of knots through the lens of topology, particularly using tools like Dehn surgery to modify and analyze these spaces. By examining knot complements, one can better understand how knots interact with their surrounding space and the implications this has for various topological properties.
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