Conway's Notation

This is one of the most important ways in which the knots can be distinguished and can also be used in the computer to generate Knots. This notation was introduced by John H.Conway. This is also been used in the knotting process in the study of DNA. In order to understand this we must first understand the term tangle. A tangle in a knot is a region in which exactly four strands going out of the circle.

Figure: The $\infty$ tangle

Figure: The '0' tangle

Figure: The 3 tangle

In order to understand this even better we have to define what is meant by positive and a negative slope. The same meaning is used here to find out which tangle has a positive slope and which has a negative slope.

You might notice that in the above figure we have some numbers written that represent the tangle. These number help us to distinguish the knots. The tangle with odd number of integers starts with a "0 tangle" and the tangle with even number of integers start with an "$\infty$ tangle". These types of tangles can be seen above. Now, you see why the two tangles mentioned and shown above are very important. They form the base for all the tangles.

Figure: -3,-2

Figure: 3,2

The integers/numbers that you see at the bottom of the tangle can be used to computer continued fractions.