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Gordon–Luecke Theorem
In mathematics, the Gordon–Luecke theorem on knot complements states that if the complements of two tame knots are homeomorphic, then the knots are equivalent. In particular, any homeomorphism between knot complements must take a meridian to a meridian. The theorem is usually stated as "knots are determined by their complements"; however this is slightly ambiguous as it considers two knots to be equivalent if there is a self-homeomorphism taking one knot to the other. Thus mirror images are neglected. Often two knots are considered equivalent if they are '' isotopic''. The correct version in this case is that if two knots have complements which are orientation-preserving homeomorphic, then they are isotopic. These results follow from the following (also called the Gordon–Luecke theorem): no nontrivial Dehn surgery on a nontrivial knot in the 3-sphere can yield the 3-sphere In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Surgery Theorem
In three-manifold, three-dimensional topology, a branch of mathematics, the cyclic surgery theorem states that, for a Compact space, compact, Connected space, connected, Orientability, orientable, Irreducibility (mathematics), irreducible three-manifold ''M'' whose boundary is a torus ''T'', if ''M'' is not a Seifert-fibered space and ''r,s'' are slopes on ''T'' such that their Dehn surgery, Dehn fillings have cyclic fundamental group, then the distance between ''r'' and ''s'' (the minimal number of times that two simple closed curves in ''T'' representing ''r'' and ''s'' must intersect) is at most 1. Consequently, there are at most three Dehn fillings of ''M'' with cyclic fundamental group. The theorem appeared in a 1987 paper written by Marc Culler, Cameron Gordon (mathematician), Cameron Gordon, John Luecke (mathematician), John Luecke and Peter Shalen.M. Culler, C. Gordon, J. Luecke, P. Shalen (1987). Dehn surgery on knots. The Annals of Mathematics (''Annals of Mathematics'') 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
