## Knot Theory

*Knot theory* is a relatively new branch of mathematical research. In the 1880s, Lord Kelvin proposed to model atoms as knots of vertices in the *ether*. Although this model was not very successful (the existence of ether was disproved in the Michelson-Morley Experiment, laying the foundations for Einstein’s *special relativity*), knot theory remained a vivid area of mathematics research, and its applications e.g. for physics are far from being given up, including brand new areas like quantum computation.

Knot theory many areas which are accessible even for undergraduates. From the physical point of view, we construct a knot by taking a segment of rope or a cord, knotting it and fusing the endpoints. A more strict definition from the mathematical point of view is that a knot is a closed loop in three-dimensional space.

### Theoretical minimum (in a nutshell)

To start studying knot theory, you should be already acquainted with the following:

- Basic concepts of combinatorics,
- The notion of equivalence classes,
- The notions of some algebraic structures, including groups and polynomials,
- Linear algebra.

### Concepts you will learn in this part of **BookofProofs**

- How to represent knots using
*Reidemeister moves*,*Gauss diagrams*, and algebraic structures called*quandles*? - What are
*knot invariants*and why they are important for the classification of knots? - How to define different equivalence classes of knot diagrams and which different knot theories result from that approach?
- Methods of knot construction.
- Which are the still unsolved problems concerning the classification of knots you are invited to do research for?

| | | | Contributors: *bookofproofs* | References: [6357]

## 1.**Definition**: Knot Diagram, Classical Crossing, Virtual Crossing

## 2.**Definition**: Reidemeister Moves, Planar Isotopy Moves, Diagrammatic Moves

## 3.**Definition**: Knot

## 4.**Definition**: Unknot

## 5.**Proposition**: Equivalent Knot Diagrams

[6357] **Dye, Heather**: “An Invitation to Knot Theory”, CRC Press, 2016

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