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A manifold with a Riemannian metric that has zero curvature is a flat manifold. The basic example is Euclidean space with the usual metric ds^2=sum_(i)dx_i^2. In fact, any ...

Two distinct knots cannot have the same exterior. Or, equivalently, a knot is completely determined by its knot exterior (Cipra 1988; Adams 1994, p. 261). The question was ...

A handle is a topological structure which can be thought of as the object produced by puncturing a surface twice, attaching a zip around each puncture travelling in opposite ...

A hole in a mathematical object is a topological structure which prevents the object from being continuously shrunk to a point. When dealing with topological spaces, a ...

The hyperbolic volume of the knot complement of a hyperbolic knot is a knot invariant. Adams (1994) lists the hyperbolic volumes for knots and links. The hyperbolic volume of ...

A quantity which remains unchanged under certain classes of transformations. Invariants are extremely useful for classifying mathematical objects because they usually reflect ...

The least genus of any Seifert surface for a given knot. The unknot is the only knot with genus 0. Usually, one denotes by g(K) the genus of the knot K. The knot genus has ...

The Riemann sphere, also called the extended complex plane, is a one-dimensional complex manifold C^* (C-star) which is the one-point compactification of the complex numbers ...

A knot K in S^3=partialD^4 is a slice knot if it bounds a disk Delta^2 in D^4 which has a tubular neighborhood Delta^2×D^2 whose intersection with S^3 is a tubular ...

Let V be a real vector space (e.g., the real continuous functions C(I) on a closed interval I, two-dimensional Euclidean space R^2, the twice differentiable real functions ...