A region in a knot or link projection plane surrounded by a circle such that the knot
or link crosses the circle exactly four times. Two tangles
are equivalent if a sequence of Reidemeister moves
can be used to transform one into the other while keeping the four string endpoints
fixed and not allowing strings to pass outside the circle.

The simplest tangles are the -tangle
and 0-tangle, shown above. A tangle with left-handed twists is called an -tangle, and one with right-handed twists is called a -tangle. By placing tangles side by side, more complicated
tangles can be built up such as (, 3, 2), etc. The link created by connecting the ends of the
tangles is now described by the sequence of tangle symbols, known as Conway's
knot notation. If tangles are multiplied by 0 and then added, the resulting tangle
symbols are separated by commas. Additional symbols which are used are the period,
colon, and asterisk.