In the "Playing with Abstractions" section of the Beyond Numbers exhibit, visitors encounter knot models, cultural string games, and even a model of "the place where a knot is not" to learn more about mathematicians' conceptions of knots.
 
 

 Mathematics as Corrections:
Students use mathematics in other curriculum areas. Students use mathematics in their daily lives. 

Patterns and Function:
Students describe, extend, analyze, and create a wide variety of patterns.

 Geometry and Spatial Sense:
Students develop spatial sense. Students recognize, describe, extend and create a wide variety of patterns.

 The Mathematical World, Symbolic Relationship:
Students learn that similar patterns may show up in many places in nature and in the things people make.

topology - the study of certain properties that remain the same when a figure is twisted, stretched, or squeezed. Some of these properties are the connections between points, the number of faces, edges, crossings, "handles", or holes, in a surface.

invariants - qualities that do not vary, or change.

knot theory - a subdiscipline of topology that focuses on the study of closed curves in three-dimensional space

genus - type, in this lesson applied to classifying surfaces

crossing - with rope knots, rope lying over or under rope 

crossing order - with rope knots, the order the path of a rope takes as it goes from crossing to crossing knot crossings the crossing order of this knot is: 

It is difficult to identify the equivalence of knots 

"...without doing some tedious checking. The problem lies in the fact that two equivalent knots may have very different planar projections. This leads to the need for characterizing knots algebraically. The idea is to associate algebraic functions with knots such that equivalent knots possess the same identical function. Such algebraic functions are the topological invariants of knots, or simply knot invariants. 

- F.Y. Wu, Reviews of Modern Physics, Vol. 64, No. 4, October 1992, American Physical Society 

The following activities serve as an introduction to knot theory and will help students appreciate the difficulty in identifying knots' invariants. 

Students will observe features of distinct knots. 

Use rope to make loose versions of four knots, roughly ten of each. Be sure that the loose ends are taped together to make a continuous form.

 Show students the "shadow" diagrams of the "trefoil" knots (both orientations) and an "unknot." They should all look the same. (You may wish to put these knots on an overhead projector to show the shadow) 

Manipulate the rope so as to reflect the "under" and "over" crossings in the diagram. Point out that the "under" or "over" crossings make these knots different. Show that, in these examples, when the loose ends are taped together, the over-under crossings cannot be undone, but the under-under crossings can.

Show students a tightly knotted left-handed trefoil knot and a loosely knotted left-handed trefoil knot. Point out that these are topologically equivalent. The crossings are the same in relation to one another. (In topology, size and shape do not matter.)

Mix the knots and give one to each student, directing them to: 

1.) Find an unused corner in the room

2.) Find people with the same knot as theirs and meet in the corner

3.) Check to see that all their knots match.

4.) When they are sure that their group is ready, fold their arms.

Tell them that you will be timing the challenge. Have them predict how long it should take. Then do the activity, as the students compete against their predicted time.

After the challenge is completed and the results checked against your code record, have students reflect on their strategies for identifying a matching knot. Have them write their strategies in narrative form.

Later you may do the same activity to see if the time improves. (It probably will.) 

Assessment: Give each student a length of rope, masking tape to connect the ends, and a copy of the diagram of classified knots below. Have teams compete with each other to form a set of knots - no two of which are the same.