Geometry and topology quiz

Question 1

Topology treats two shapes as the same when one can be turned into the other by

Accepted Answer

stretching and bending, without tearing or gluing. Topology deliberately forgets distance and angle. A coffee cup and a doughnut are the same because each has one hole; a sphere is different because it has none.

Question 2

Every point of a manifold has a neighbourhood resembling Euclidean space of

Accepted Answer

one fixed dimension. The fixed dimension is part of the definition. A space that is two-dimensional in one region and three-dimensional in another satisfies the local condition and is not a manifold.

Question 3

The genus of a torus is

Accepted Answer

1. Genus counts holes, and it classifies closed orientable surfaces completely: two are topologically identical exactly when their genus agrees.

Question 4

For any convex polyhedron, vertices minus edges plus faces equals

Accepted Answer

2. Every convex polyhedron is topologically a sphere, so the quantity belongs to the shape rather than the subdivision. A torus gives 0 instead.

Question 5

Knot theory studies loops that are

Accepted Answer

closed, with the ends joined. An open piece of string can always be untangled, so the mathematics only becomes interesting once the ends are joined.

Question 6

To prove two knots are genuinely different, you need

Accepted Answer

an invariant that deformation cannot change. Failing to find a deformation proves nothing, since you may simply not have found it. An invariant is a quantity preserved by every deformation, so two knots differing in one cannot be the same knot.