Topology Intuition Math Example 4
Follow the full solution, then compare it with the other examples linked below.
Example 4
mediumA topologist says that the number of times a closed curve crosses itself is a topological property that distinguishes curves. A circle crosses itself 0 times. A figure-8 crosses itself once. Can you continuously deform a figure-8 into a circle without lifting it from the plane?
Solution
- 1 Step 1: The crossing number is a topological invariant for curves in the plane — it cannot change under continuous deformation without passing the curve through itself.
- 2 Step 2: A circle has 0 crossings; a figure-8 has 1 crossing. These are different invariants.
- 3 Step 3: To go from 1 crossing to 0 crossings, you would need to pass one part of the curve through another, which is not allowed in a continuous deformation in the plane.
Answer
No, a figure-8 cannot be continuously deformed into a circle in the plane.
The self-intersection number of a curve in the plane is preserved by continuous deformations (homeomorphisms of the plane). Since the circle has 0 self-intersections and the figure-8 has 1, they are topologically distinct as plane curves, and no continuous deformation in the plane can transform one into the other.
About Topology Intuition
Properties of shapes that are preserved under continuous deformation (stretching, bending, and twisting, but not tearing or gluing). Topology studies what remains the same when you treat shapes as if they were made of infinitely stretchable rubber.
Learn more about Topology Intuition →More Topology Intuition Examples
Example 1 medium
A rubber band is shaped like a circle. If you stretch and reshape it (without tearing or gluing), ca
Example 2 hardA coffee mug and a donut (torus) are famously topologically equivalent. A sphere and a donut are not
Example 3 easyWhich pairs of shapes are topologically equivalent (same number of holes)? (a) Triangle and circle.