Knot Theory Module

Anticipated Learning Outcomes

About Knots

Give an informal definition of a knot.

Manipulate physical representations of a knot.

Form a repertoire of knots and links.

Give a definition of a knot or link projection.

Distinguish between a knot and its unlimited number of projections.

Draw, step-by-step, equivalent projections of a knot.

Calculate the linking number of a two-component link.

Understand the meaning and use of knot characteristics to identify and differentiate between knots and how these characteristics are knot invariants.

Describe what it means for two knots to be equivalent. Describe how this is different from Reidemeister equivalence.

Describe the relationship of physical moves to Reidemeister moves.

Understand that Reidemeister moves are helpful as theoretical tool, not so useful for actually simplifying a given projection.

Know definition of an alternating knot. Distinguish alternating knots projections from non-alternating knot projections.

Describe the connection between alternating projections and crossing numbers.

State the first Tait conjecture and know it stood for about 100 years until proven using the Jones polynomial.

Understand the major problem in knot theory.

Understand the relationship of invariants to the major problem.

Understand that mathematicians don't have a complete invariant. In other words, for every invariant we know of, there are two different knots which have the same value for that invariant.

Lake and Island polynomial is not a knot invariant. Jones polynomial is.

The Jones polynomial of left and right trefoil are different.

The Jones polynomial of trefoil, figure 8, Listing's knot and star knot are all different from one another.

 

About Functions

Develop an expanded notion of function to include arbitrary sets, different names for the same element in the domain, uniqueness of image.

Relate the concept of "knot invariant" to the concept of function.

Relate the concept of "complete invariant" to the concept of function.

Other mathematical habits and skills

Visualization

Problem-solving

Conjecture and proof