$\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{{\not{\mid}}} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&}$

## Section2.1Knots Overview

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.

### Subsection2.1.1Objectives

1. Determine and contrast several ways to notate (tabulate) a knot using its crossings.
2. Use numerical invariants for knots including the unknotting, bridge, and crossing numbers, to investigate relationships among classes of knots.

### Subsection2.1.2References

[4]

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.
[7]

Rolfsen, D. (1990). Knots and Links. Corrected reprint of the 1976 original. Mathematics Lecture Series (7). American Mathematical Society. Chapter 3.