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Truth conditions are the circumstances under which a statement is considered true, serving as a fundamental aspect of semantic theory in understanding meaning. They help distinguish between truth-value and meaning, emphasizing the role of context and interpretation in linguistic analysis.
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Knot theory is a branch of topology that studies mathematical knots, which are embeddings of a circle in 3-dimensional space, focusing on their properties and classifications. It has applications in various fields, including biology, chemistry, and physics, where it helps in understanding the structure of DNA, molecular compounds, and the behavior of physical systems.
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The three-sphere is a higher-dimensional analogue of a sphere, specifically existing in four-dimensional Euclidean space. It is a smooth, compact, and simply connected manifold, often used in the study of topology and geometry to explore complex spatial relationships and properties.
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A topological space is a fundamental concept in mathematics that generalizes the notion of geometric spaces, allowing for the definition of continuity, convergence, and boundary without requiring a specific notion of distance. It is defined by a set of points and a topology, which is a collection of open sets that satisfy certain axioms regarding unions, intersections, and the inclusion of the entire set and the empty set.
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A manifold is a topological space that locally resembles Euclidean space, allowing for the application of calculus and other mathematical tools. Manifolds are fundamental in mathematics and physics, providing the framework for understanding complex structures like curves, surfaces, and higher-dimensional spaces.
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The fundamental group is an algebraic structure that captures the topological essence of a space by describing the loops in the space up to continuous deformation. It is a powerful invariant in topology that helps distinguish between different topological spaces by examining the equivalence classes of loops based at a point.
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Dehn surgery is a technique in 3-dimensional topology that involves modifying a manifold by removing a tubular neighborhood of a knot and gluing it back in a different way, which can yield diverse and exotic manifolds. This method is crucial for understanding and classifying 3-manifolds, as it provides a means to construct manifold structures from simpler components.
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Hyperbolic geometry is a non-Euclidean geometry where the parallel postulate does not hold, allowing for multiple parallel lines through a given point not on a line. It features unique properties such as the sum of angles in a triangle being less than 180 degrees and the concept of hyperbolic space, which models hyperbolic surfaces and spaces with constant negative curvature.
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The Alexander Polynomial is an invariant of knots and links in three-dimensional space that provides insights into their topological properties. It is a powerful tool in knot theory, useful for distinguishing non-equivalent knots and understanding their symmetries.
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A Seifert surface is a crucial construct in topology, providing an orientable surface whose boundary is a given knot or link in three-dimensional space. This concept illuminates the link between knot theory and surface topology, facilitating the study of knot invariants and the application of knot theory in higher dimensions.
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In graph theory, the link complement of a given graph is a graph that contains all the edges not present in the original graph, while sharing the same set of vertices. This operation is useful in network analysis and theoretical computer science for exploring properties like connectivity and independence in complementary structures.
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Knot classification is a branch of topology that deals with the categorization and study of knots, which are embeddings of circles in 3-dimensional space, up to continuous deformations known as isotopies. This field seeks to understand the properties and invariants that distinguish different knots, aiding in applications across mathematics, physics, and biology.
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A fibered knot is a type of knot in three-dimensional space that has a special kind of structure, allowing it to be the boundary of a surface called a fiber surface, which fibers the knot complement in the three-sphere. This characteristic makes fibered knots important in the study of 3-manifolds and knot theory, as they provide insight into the topology and geometry of the spaces they inhabit.
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A knot group is the fundamental group of the complement of a knot in three-dimensional space, providing a topological invariant that distinguishes different knots. It encapsulates the algebraic structure of how loops around the knot can be deformed, offering insights into the knot's properties and classification.
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Knot invariants are properties of knots that remain unchanged under ambient isotopies, providing a way to distinguish between different knots. They are crucial in the study of knot theory, a branch of topology, and are used to classify and understand the complex interactions of knots in three-dimensional space.
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A prime knot is a non-trivial knot that cannot be represented as the knot sum of two non-trivial knots, essentially serving as the building blocks for all other knots. Understanding prime knots is crucial in knot theory as they help in classifying and distinguishing different types of knots through various knot invariants.
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Satellite knots are a type of knot in three-dimensional space formed by taking a nontrivial knot, known as the companion knot, and tying it around a torus that is itself knotted in a particular way. They are significant in the study of knot theory because they help in understanding the properties of more complex knots through their construction from simpler ones.
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