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

## Section1.1Tangles Overview

As we discovered in our first class, crossings are one of the first ways for us to understand the connections between knots and algebra: somehow, if we can say "enough" about how a strand crosses itself, we can characterize the essential nature of a knot.

So we'll begin by focusing as much as possible only on crossings, by studying objects known as tangles, in which crossings are created between two strands by twisting up their endpoints.

### Subsection1.1.1Objectives

1. Discover how different types of twists on a tangle determine its tangle number.
2. Argue for why the arithmetic of the rational numbers makes certain relationships among tangle numbers necessary.
3. Calculate the fraction of a rational tangle in two different ways, and argue for why the fraction is an invariant of rational tangles.

### Subsection1.1.2References



Kauffman, L. H., & Lambropoulou, S. (2004). On the classification of rational tangles. Advances in Applied Mathematics, 33(2), 199-237. Available on arXiv at http://arxiv.org/pdf/math/0311499.pdf.