Editing Knot theory (section)

===Reidemeister moves===
{{main|Reidemeister move}}
In 1927, working with this diagrammatic form of knots, [[James Waddell Alexander II|J. W. Alexander]] and [[Garland Baird Briggs]], and independently [[Kurt Reidemeister]], demonstrated that two knot diagrams belonging to the same knot can be related by a sequence of three kinds of moves on the diagram, shown below. These operations, now called the ''Reidemeister moves'', are:

{{Ordered list|list-style-type=upper-Roman
|Twist and untwist in either direction.
|Move one strand completely over another.
|Move a strand completely over or under a crossing.
}}

{| align="center" style="text-align:center"
|+ '''Reidemeister moves'''
|- style="padding:1em"
| [[File:Reidemeister move 1.png|130px]] [[File:Frame left.png]] || [[File:Reidemeister move 2.png|210px]]
|-
! Type I !! Type II
|- style="padding:1em"
| colspan="2" | [[File:Reidemeister move 3.png|360px]]
|-
! colspan="2" | Type III
|}

The proof that diagrams of equivalent knots are connected by Reidemeister moves relies on an analysis of what happens under the planar projection of the movement taking one knot to another.  The movement can be arranged so that almost all of the time the projection will be a knot diagram, except at finitely many times when an "event" or "catastrophe" occurs, such as when more than two strands cross at a point or multiple strands become tangent at a point.  A close inspection will show that complicated events can be eliminated, leaving only the simplest events: (1) a "kink" forming or being straightened out;  (2) two strands becoming tangent at a point and passing through; and (3) three strands crossing at a point.  These are precisely the Reidemeister moves {{Harv|Sossinsky|2002| loc=ch. 3}} {{Harv|Lickorish|1997| loc=ch. 1}}.