Editing Knot theory (section)

==Knot invariants==
[[File:Figure eight knot complement.jpg|thumb|right| text-bottom |280px|  A 3D print depicting the complement of the figure eight knot<br>by François Guéritaud, Saul Schleimer, and [[Henry Segerman]]]]
{{main|Knot invariant}}
A knot invariant is a "quantity" that is the same for equivalent knots {{Harv|Adams|2004}} {{Harv|Lickorish|1997}} {{Harv|Rolfsen|1976}}.  For example, if the invariant is computed from a knot diagram, it should give the same value for two knot diagrams representing equivalent knots.  An invariant may take the same value on two different knots, so by itself may be incapable of distinguishing all knots. An elementary invariant is [[tricolorability]].

"Classical" knot invariants include the [[knot group]], which is the [[fundamental group]] of the [[knot complement]], and the [[Alexander polynomial]], which can be computed from the Alexander invariant, a module constructed from the infinite cyclic cover of the knot complement {{Harv|Lickorish|1997}}{{Harv|Rolfsen|1976}}. In the late 20th century, invariants such as "quantum" knot polynomials, [[Vassiliev invariant]]s and hyperbolic invariants were discovered. These aforementioned invariants are only the tip of the iceberg of modern knot theory.

===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}}.

===Hyperbolic invariants===
[[William Thurston]] proved many knots are [[hyperbolic knot]]s, meaning that the [[knot complement]] (i.e., the set of points of 3-space not on the knot) admits a geometric structure, in particular that of [[hyperbolic geometry]]. The hyperbolic structure depends only on the knot so any quantity computed from the hyperbolic structure is then a knot invariant {{Harv|Adams|2004}}.

{{multiple image
 | align = right
 | total_width = 320
 | image1 = BorromeanRings.svg
 | width1 = 626 | height1 = 600
 | caption1 = The [[Borromean rings]] are a link with the property that removing one ring unlinks the others.
 | image2 = SnapPea-horocusp_view.png
 | width2 = 560 | height2 = 416
 | caption2 = [[SnapPea]]'s cusp view: the [[Borromean rings]] complement from the perspective of an inhabitant living near the red component.
}}

Geometry lets us visualize what the inside of a knot or link complement looks like by imagining light rays as traveling along the [[geodesic]]s of the geometry. An example is provided by the picture of the complement of the [[Borromean rings]]. The inhabitant of this link complement is viewing the space from near the red component. The balls in the picture are views of [[horoball]] neighborhoods of the link. By thickening the link in a standard way, the horoball neighborhoods of the link components are obtained. Even though the boundary of a neighborhood is a torus, when viewed from inside the link complement, it looks like a sphere. Each link component shows up as infinitely many spheres (of one color)  as there are infinitely many light rays from the observer to the link component. The fundamental parallelogram (which is indicated in the picture), tiles both vertically and horizontally and shows how to extend the pattern of spheres infinitely.

This pattern, the horoball pattern, is itself a useful invariant. Other hyperbolic invariants include the shape of the fundamental parallelogram, length of shortest geodesic, and volume. Modern knot and link tabulation efforts have utilized these invariants effectively. Fast computers and clever methods of obtaining these invariants make calculating these invariants, in practice, a simple task {{Harv|Adams|Hildebrand|Weeks|1991}}.