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The four-color theorem states that any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary (other than a single point) do not ...

A perfect magic cube is a magic cube for which the rows, columns, pillars, space diagonals, and diagonals of each n×n orthogonal slice sum to the same number (i.e., the magic ...

The Cassini ovals are a family of quartic curves, also called Cassini ellipses, described by a point such that the product of its distances from two fixed points a distance ...

The toroidal crossing number cr_(1)(G) of a graph G is the minimum number of crossings with which G can be drawn on a torus. A planar graph has toroidal crossing number 0, ...

Letting Lk be the linking number of the two components of a ribbon, Tw be the twist, and Wr be the writhe, then Lk(K)=Tw(K)+Wr(K). (Adams 1994, p. 187).

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

In the usual diagram of inclusion homomorphisms, if the upper two maps are injective, then so are the other two. More formally, consider a space X which is expressible as the ...

The prime link 02-0201 which has Jones polynomial V(t)=-t-t^(-1) and HOMFLY polynomial P(z,alpha)=z^(-1)(alpha^(-1)-alpha^(-3))+zalpha^(-1). It has braid word sigma_1^2.

The unlink, also called the trivial link, of n components consist of n disjoint circles in a plane (Rolfsen 1976, p. 65).

Every planar graph (i.e., graph with graph genus 0) has an embedding on a torus. In contrast, toroidal graphs are embeddable on the torus, but not in the plane, i.e., they ...