Seifert Surfaces
Description
Using knots made out of wire, students will
see what happens when they dip them into bubble-making solutions. They
will experience the idea of a surface, and
they will experience how the study of knots can be taken in amazing
directions.
Materials
Instructions
Ideas for discussion
- trefoil knots made out of
wire that can be dipped to make bubbles.
- Pans or trays of detergent or other bubble solution. (Adding a
little glycerin to ordinary dish detergent makes very sturdy
bubbles that last long enough to be examined carefully.)
- Additional wire for making other knot-shaped bubble-makers.
- Have students dip trefoil knots into
the bubble solution
and examine carefully the way the soap film clings to the wire. Ask
students to describe what they see.
- The form of the stretched and twisted soap film is called a Seifert Surface . Notice that it has two
sides, a "front"
and a "back". When you hold the surface up, which parts face you? You
can draw an accurate representation of the surface by using two
colors, one for each side.
- Ask students to choose another knot that they would like to
twist from the wire and examine the soap film that it will make.
Before dipping the knot into the bubble mixture, have students predict
the surface that they think will form.
- Experiment with other knot-shaped bubble makers and catalog the
surfaces that form.
- Describe what happens when various knots are dipped. Why do you
think this occurs?
- Does the same surface always form when knots are dipped? Is there a
way you can be sure that no surfaces other than the ones you have
discovered will ever form when a knot is dipped?
- How can you account for the things that you observe?
