Up: Winding Numbers

Exploring Winding Numbers and Homotopies

Changing the sliders below applies a homotopy to the colored curve. The winding number of the curve only changes when the homotopy pushes the curve through the origin.

When slider a1 = 1 and the others are zero, the colored curve is in a standard position for a curve with winding number 1. When a2 = 1, and the others are zero, the winding number is 2, and similarly for a3. The standard curve with winding number 3 is indicated in black.

Given any initial configuration there is a sequence of moves (i.e. a homotopy) taking the colored curve to one of the standard ones. Can you find it?

As you play with the sliders, you can use the three colored segments on the curve to keep track of the winding number. Follow the curve in the "red" -> "green" -> "blue" direction, and count the number of times you encircle the origin. Observe how the winding number changes as you pass over the origin.

Acknowledgements: George Peschke (gepe@jazz.math.ualberta.ca) at the University of Alberta suggested the idea for this applet to me. He is in the process of developing extensive additional online materials.

Up: Winding Numbers
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Comments to: rminer@geom.umn.edu
Created: Fri Sep 8 11:39:00 1995 --- Last modified: Wed Apr 17 17:15:53 1996