Editing Knot theory (section)

===Knot polynomials===
{{main|Knot polynomial}}
A knot polynomial is a [[knot invariant]] that is a [[polynomial]]. Well-known examples include the [[Jones polynomial]], the [[Alexander polynomial]], and the [[Kauffman polynomial]]. A variant of the Alexander polynomial, the [[Alexander–Conway polynomial]], is a polynomial in the variable ''z'' with [[integer]] coefficients {{Harv|Lickorish|1997}}.

The Alexander–Conway polynomial is actually defined in terms of [[link (knot theory)|links]], which consist of one or more knots entangled with each other.  The concepts explained above for knots, e.g. diagrams and Reidemeister moves, also hold for links.

Consider an oriented link diagram, ''i.e.'' one in which every component of the link has a preferred direction indicated by an arrow. For a given crossing of the diagram, let <math>L_+, L_-, L_0</math> be the oriented link diagrams resulting from changing the diagram as indicated in the figure:  [[File:Skein (HOMFLY).svg|200px|center]]

The original diagram might be either <math>L_+</math> or <math>L_-</math>, depending on the chosen crossing's configuration. Then the Alexander–Conway polynomial, <math>C(z)</math>, is recursively defined according to the rules:

* <math>C(O) = 1</math> (where <math>O</math> is any diagram of the [[unknot]])
* <math>C(L_+) = C(L_-) + z C(L_0).</math>

The second rule is what is often referred to as a [[skein relation]]. To check that these rules give an invariant of an oriented link, one should determine that the polynomial does not change under the three Reidemeister moves. Many important knot polynomials can be defined in this way.

The following is an example of a typical computation using a skein relation. It computes the Alexander–Conway polynomial of the [[trefoil knot]]. The yellow patches indicate where the relation is applied.

:''C''([[File:skein-relation-trefoil-plus-sm.png]])&nbsp;=&nbsp;''C''([[File:skein-relation-trefoil-minus-sm.png]])&nbsp;+&nbsp;''z'' ''C''([[File:skein-relation-trefoil-zero-sm.png]])

gives the unknot and the [[Hopf link]]. Applying the relation to the Hopf link where indicated,

:''C''([[File:skein-relation-link22-plus-sm.png]]) = ''C''([[File:skein-relation-link22-minus-sm.png]]) + ''z'' ''C''([[File:skein-relation-link22-zero-sm.png]])

gives a link deformable to one with 0 crossings (it is actually the [[unlink]] of two components) and an unknot. The unlink takes a bit of sneakiness:

:''C''([[File:skein-relation-link20-plus-sm.png]]) = ''C''([[File:skein-relation-link20-minus-sm.png]]) + ''z''  ''C''([[File:skein-relation-link20-zero-sm.png]])

which implies that ''C''(unlink of two components) = 0, since the first two polynomials are of the unknot and thus equal.

Putting all this together will show:

:<math>C(\mathrm{trefoil}) = 1 + z(0 + z) = 1 + z^2</math>

Since the Alexander–Conway polynomial is a knot invariant, this shows that the trefoil is not equivalent to the unknot. So the trefoil really is "knotted".

<gallery widths="80px" heights="80px" align="right">
Image:Trefoil knot left.svg|The left-handed trefoil knot.
Image:TrefoilKnot_01.svg|The right-handed trefoil knot.
</gallery>
Actually, there are two trefoil knots, called the right and left-handed trefoils, which are [[chiral knot|mirror images]] of each other (take a diagram of the trefoil given above and change each crossing to the other way to get the mirror image). These are not equivalent to each other, meaning that they are not amphichiral.  This was shown by [[Max Dehn]], before the invention of knot polynomials, using [[group theory|group theoretical]] methods {{Harv|Dehn|1914}}. But the Alexander–Conway polynomial of each kind of trefoil will be the same, as can be seen by going through the computation above with the mirror image. The ''Jones'' polynomial can in fact distinguish between the left- and right-handed trefoil knots {{Harv|Lickorish|1997}}.