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

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.


  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.


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