Examples for

Topology

Topology is the study of the properties of space preserved under continuous deformations, including twisting and stretching. Topology originated from the study of geometry and set theory and has many branches today. Wolfram|Alpha has the knowledge and ability to handle multiple topological topics, like computing the homotopy groups of an n-sphere (algebraic topology) or visualizing a knot (geometric topology).

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Real Intervals

Compute properties for one real interval or compare multiple intervals.

Compute properties of an interval of real numbers:

[3,8)

Compare intervals:

intervals [0,1), (1,oo)

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Knot Theory

Find and visualize knots, either by specifying a common name or using knot notation.

Compute properties of a knot:

trefoil knot

Specify a knot using Alexander–Briggs notation:

8_1 knot

Compare several knots:

Stevedore knot, figure eight knot

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Homotopy Groups of Spheres

Compute the homotopy groups of an n-sphere.

Compute homotopy groups of a sphere:

homotopy 3-sphere

Compute a specified homotopy group:

pi_5(S^2)

pi_719(S^700)

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