In mathematics, the knot complement of a tame knot K is the set-theoretic complement of the interior of the embedding of a solid torus into the 3-sphere. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a knot is an Embedding of a Circle in 3-dimensional Euclidean space, R 3 considered up to continuous deformations In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation In Mathematics, a solid torus is a Topological space Homeomorphic to S^1 \times D^2 i In Mathematics, a 3-sphere is a higher-dimensional analogue of a Sphere. This solid torus is a thickened neighborhood of K. Note that the knot complement is a compact 3-manifold with boundary homeomorphic to a torus. In Mathematics, a 3-manifold is a 3-dimensional Manifold. The topological Piecewise-linear, and smooth categories are all equivalent in three dimensions In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar Sometimes "knot complement" means the complement in the 3-sphere of a knot (whether tame or not), in which case the knot complement is not compact. In Mathematics, a 3-sphere is a higher-dimensional analogue of a Sphere. Context is needed to determine the usage. There are analogous definitions of link complement. In Mathematics, a link is a collection of knots which do not intersect but which may be linked (or knotted together
Many knot invariants, such as the knot group, are really invariants of the complement of the knot. In the mathematical field of Knot theory, a knot invariant is a quantity (in a broad sense defined for each knot which is the same for equivalent knots In Mathematics, a knot is an Embedding of a Circle into 3-dimensional Euclidean space. This is not a disadvantage because the Gordon-Luecke theorem states that a knot is determined by its complement. In Mathematics, the Gordon-Luecke theorem on Knot complements states that every Homeomorphism between two complements of knots in the 3-sphere That is if K and K′ are two knots with homeomorphic complements then there is a homeomorphism of the 3-sphere taking one knot to the other. Topological equivalence redirects here see also Topological equivalence (dynamical systems.
References
- C. Gordon and J. Luecke, "Knots are Determined by their Complements", J. Amer. Math. Soc., 2 (1989), 371–415. The Journal of the American Mathematical Society, often referred to by its acronym JAMS, is a mathematics journal published quarterly by the American Mathematical Society