2: Knots and Numbers
- Determine and contrast several ways to notate (tabulate) a knot using its crossings.
- Use numerical invariants for knots including the unknotting, bridge, and crossing numbers, to investigate relationships among classes of knots.
Studying rational tangles was a way to focus in a limited fashion on how crossings interact with one another to build intricate local structures that define a knot. But as an invariant for knots, the tangle number isn't perfect: it's most useful for rational knots, and even then, it can be challenging to rearrange a knot diagram into a twist-form rational tangle.
What we'd like instead are more global invariants that work for knots, invariants that capture the whole structure of the topology without relying upon making a specific set of choices along the way. This will come at the cost of needing invariants capable of conveying more algebraic information than a single rational number does: polynomials on one hand, and algebraic groups on the other.
References
- Adams, C. C. (2004). The Knot Book: an elementary introduction to the mathematical theory of knots. American Mathematical Society, ISBN 0-8218-3678-1. Chapters 2 and 3.
- Austin, D. (2016). Knot quandaries quelled by quandles. American Mathematical Society Feature Column, accessed at http://www.ams.org/publicoutreach/feature-column/fc-2016-03 .
- Portnoy, N. and Mattman, T. (Undated). Knot Theory for Preservice and Practicing Secondary Mathematics Teachers. Accessed at http://www.csuchico.edu/math/mattman/NSF.html .
- Rolfsen, D. (1990). Knots and Li nks. Corrected reprint of the 1976 original. Mathematics Lecture Series (7). American Mathematical Society. Chapter 3