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A 3-manifold is a space that locally resembles Euclidean 3-dimensional space, meaning each point has a neighborhood that looks like the Euclidean space R^3. Understanding 3-manifolds is crucial in topology and geometry, particularly in the study of the universe's shape and the field of 3-dimensional topology, where they serve as the primary objects of study.
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The Kauffman bracket is a polynomial invariant of framed links in three-dimensional space, playing a crucial role in knot theory and low-dimensional topology. It serves as a precursor to the Jones polynomial, providing a combinatorial method for evaluating link diagrams through a recursive skein relation.
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The Kauffman polynomial is an invariant used in knot theory, which generalizes the Jones polynomial by incorporating two variables. It provides a powerful tool for distinguishing between different links and knots, offering deeper insights into the topology of three-dimensional spaces.
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