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A torispherical dome is the surface obtained from the intersection of a spherical cap with a tangent torus, as illustrated above. The radius of the sphere R is called the ...

An n-Hadamard graph is a graph on 4n vertices defined in terms of a Hadamard matrix H_n=(h)_(ij) as follows. Define 4n symbols r_i^+, r_i^-, c_i^+, and c_i^-, where r stands ...

The homotopy groups generalize the fundamental group to maps from higher dimensional spheres, instead of from the circle. The nth homotopy group of a topological space X is ...

Smale (1958) proved that it is mathematically possible to turn a sphere inside-out without introducing a sharp crease at any point. This means there is a regular homotopy ...

Any nondegenerate closed space curve may be nondegenerately deformed into either of the two curves illustrated above. Neither of these can be nondegenerately transformed into ...

The inversion of a horn torus. If the inversion center lies on the torus, then the horn cyclide degenerates to a parabolic horn cyclide.

Four circles may be drawn through an arbitrary point P on a torus. The first two circles are obvious: one is in the plane of the torus and the second perpendicular to it. The ...

A type II Markov move.

The Reidemeister move of type I.

The Conway polynomial del _L(x), sometimes known as the Conway-Alexander polynomial, is a modified version of the Alexander polynomial Delta_L(x) that was formulated by J. H. ...