Editing Knot theory (section)

==Knot diagrams<!--[[Knot diagram]] links directly here, [[MOS:HEAD]]-->==
[[File:Tenfold Knottiness, plate IX.png|thumb|Tenfold Knottiness, plate IX, from [[Peter Guthrie Tait]]'s article "On Knots", 1884]]
A useful way to visualise and manipulate knots is to project the knot onto a plane—think of the knot casting a shadow on the wall. A small change in the direction of projection will ensure that it is [[Injective function|one-to-one]] except at the double points, called ''crossings'', where the "shadow" of the knot crosses itself once transversely {{Harv|Rolfsen|1976}}. At each crossing, to be able to recreate the original knot, the over-strand must be distinguished from the under-strand. This is often done by creating a break in the strand going underneath. The resulting diagram is an [[immersed plane curve]] with the additional data of which strand is over and which is under at each crossing. (These diagrams are called '''knot diagrams''' when they represent a [[Knot (mathematics)|knot]] and '''link diagrams''' when they represent a [[Link (knot theory)|link]].) Analogously, knotted surfaces in 4-space can be related to [[immersed surface]]s in 3-space.

A '''reduced diagram''' is a knot diagram in which there are no '''reducible crossings''' (also '''nugatory''' or '''removable crossings'''), or in which all of the reducible crossings have been removed.{{sfn|Weisstein|2013}}{{sfn|Weisstein|2013a}} A [[petal projection]] is a type of projection in which, instead of forming double points, all strands of the knot meet at a single crossing point, connected to it by loops forming non-nested "petals".{{sfn|Adams|Crawford|DeMeo|Landry|2015}}

===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}}.