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To rearrange without cutting or pasting.

The problem of deciding if two knots in three-space are equivalent such that one can be continuously deformed into another.

A knot diagram which does not specify whether crossings are under- or overcrossings.

A locally finite space is one for which every point of a given space has a neighborhood that meets only finitely many elements of any cover.

Topological lower bounds in terms of Betti numbers for the number of critical points form a smooth function on a smooth manifold.

An unknot which can only be unknotted by first increasing the number of crossings.

A vector space possessing a norm.

Two subspaces S_1 and S_2 of R^n are said to be orthogonal if the dot product v_1·v_2=0 for all vectors v_1 in S_1 and all v_2 in S_2.

A braid in which strands are intertwined in the center and are free in "handles" on either side of the diagram.

A smooth manifold with a Poisson bracket defined on its function space.