View source for Figure-eight knot (mathematics)

{{short description|Unique knot with a crossing number of four}}
{{About|the mathematical concept|the knot|Figure-eight knot|other uses|Figure 8 (disambiguation){{!}}Figure 8}}
{{Infobox knot theory
| name=              Figure-eight knot
| practical name=    Figure-eight knot
| image=             Blue Figure-Eight Knot.png
| caption=          
| arf invariant=     1
| braid length=      4
| braid number=      3
| bridge number=     2
| crosscap number=   2
| crossing number=   4
| genus=             1
| hyperbolic volume= 2.02988
| linking number=   
| stick number=      7
| unknotting number= 1
| conway_notation=   [22]
| ab_notation=       4<sub>1</sub>
| dowker notation=   4, 6, 8, 2
| thistlethwaite=   
| other=            
 | alternating=      alternating
 | class=            hyperbolic
 | fibered=          fibered
 | prime=            prime
 | slice=           
 | symmetry=         fully amphichiral
 | tricolorable=    
 | twist=            twist
| last crossing=     3
| last order=        1
| next crossing=     5
| next order=        1
}}
[[File:Figure8knot-mathematical-knot-theory.svg|100px|thumb|Figure-eight knot of practical knot-tying, with ends joined]]

In [[knot theory]], a '''figure-eight knot''' (also called '''Listing's knot'''<ref>{{Cite web|title=Listing knot - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Listing_knot|access-date=2020-06-25|website=encyclopediaofmath.org}}</ref>) is the unique knot with a [[crossing number (knot theory)|crossing number]] of four.  This makes it the knot with the third-smallest possible crossing number, after the [[unknot]] and the
[[trefoil knot]]. The figure-eight knot is a [[prime knot]].

== Origin of name ==
The name is given because tying a normal [[figure-eight knot]] in a rope and then joining the ends together, in the most natural way, gives a model of the mathematical knot.

== Description ==

A simple parametric representation of the figure-eight knot is as the set of all points (''x'',''y'',''z'') where

:<math> \begin{align}
          x & = \left(2 + \cos{(2t)} \right) \cos{(3t)} \\
          y & = \left(2 + \cos{(2t)} \right) \sin{(3t)} \\
          z & = \sin{(4t)}
        \end{align}    </math>

for ''t'' varying over the real numbers (see 2D visual realization at bottom right).

The figure-eight knot is [[Prime knot|prime]], [[alternating knot|alternating]], [[rational knot|rational]] with an associated value
of 5/3,<ref>{{cite web |last1=Gruber |first1=Hermann |title=Rational Knots with 4 crossings |url=http://home.in.tum.de/~gruberh/ |website=Rational Knots database |access-date=5 May 2022 |archive-url=https://web.archive.org/web/20060209202316/http://home.in.tum.de/~gruberh/ |archive-date=2006-02-09 |url-status=dead}}</ref> and is [[Chiral knot|achiral]]. The figure-eight knot is also a [[fibered knot]]. This follows from other, less simple (but very interesting) representations of the knot: 

(1) It is a ''homogeneous''<ref group="note">A braid is called homogeneous if every
generator <math>\sigma_i</math> either occurs always with positive or always with negative sign.</ref> [[closed braid]] (namely, the closure of the 3-string braid σ<sub>1</sub>σ<sub>2</sub><sup>−1</sup>σ<sub>1</sub>σ<sub>2</sub><sup>−1</sup>), and a theorem of [[John Stallings]] shows that any closed homogeneous braid is fibered. 

(2) It is the link at (0,0,0,0) of an [[isolated critical point]] of a real-polynomial map <var>F</var>: '''R'''<sup>4</sup>→'''R'''<sup>2</sup>, so (according to a theorem of [[John Milnor]]) the [[Milnor map]] of <var>F</var> is actually a fibration.  [[Bernard Perron]] found the first such <var>F</var> for this knot, namely,

: <math>F(x, y, z, t)=G(x, y, z^2-t^2, 2zt),\,\!</math>

where

: <math>\begin{align}
                G(x,y,z,t)=\ & (z(x^2+y^2+z^2+t^2)+x (6x^2-2y^2-2z^2-2t^2), \\
                             & \ t x \sqrt{2}+y (6x^2-2y^2-2z^2-2t^2)).
          \end{align}</math>

== Mathematical properties ==

The figure-eight knot has played an important role historically (and continues to do so) in the theory of [[3-manifold]]s.  Sometime in the mid-to-late 1970s, [[William Thurston]] showed that the figure-eight was [[hyperbolic knot|hyperbolic]], by [[manifold decomposition|decomposing]] its [[knot complement|complement]] into two [[ideal point|ideal]] [[hyperbolic geometry|hyperbolic]] [[tetrahedra]].  (Robert Riley and Troels Jørgensen, working independently of each other, had earlier shown that the figure-eight knot was hyperbolic by other means.) This construction, new at the time, led him to many powerful results and methods.  For example, he was able to show that all but ten [[Dehn surgery|Dehn surgeries]] on the figure-eight knot resulted in non-[[Haken manifold|Haken]], non-[[Seifert fiber space|Seifert-fibered]] [[Prime decomposition (3-manifold)|irreducible]] 3-manifolds; these were the first such examples.  Many more have been discovered by generalizing Thurston's construction to other knots and links.

The figure-eight knot is also the hyperbolic knot whose complement has the smallest possible [[hyperbolic volume (knot)|volume]], <math>6\Lambda(\pi/3) \approx 2.02988...</math> {{OEIS|A091518}}, where <math>\Lambda</math> is the [[Lobachevsky function]].<ref>{{Citation |author=William Thurston |author-link=William Thurston |date=March 2002 |title=The Geometry and Topology of Three-Manifolds |url=http://library.msri.org/books/gt3m/ |chapter=7. Computation of volume |chapter-url=http://library.msri.org/books/gt3m/PDF/7.pdf |page=165 |access-date=2020-10-19 |archive-date=2020-07-27 |archive-url=https://web.archive.org/web/20200727020107/http://library.msri.org/books/gt3m/ |url-status=dead }}</ref> From this perspective, the figure-eight knot can be considered the simplest hyperbolic knot.   The figure eight knot complement is a [[covering space|double-cover]] of the [[Gieseking manifold]], which has the smallest volume among non-compact hyperbolic 3-manifolds.

The figure-eight knot and the [[(−2,3,7) pretzel knot]] are the only two hyperbolic knots known to have more than 6 ''exceptional surgeries'', Dehn surgeries resulting in a non-hyperbolic 3-manifold; they have 10 and 7, respectively.  A theorem of [[Marc Lackenby|Lackenby]] and Meyerhoff, whose proof relies on the [[geometrization conjecture]] and [[computer-assisted proof|computer assistance]], holds that 10 is the largest possible number of exceptional surgeries of any hyperbolic knot. However, it is not currently known whether the figure-eight knot is the only one that achieves the bound of 10.  A well-known conjecture is that the bound (except for the two knots mentioned) is 6.
{|
|-
|[[File:Figure8knot-math-square.svg|thumb|Simple squared depiction of figure-eight configuration.]] || [[File:Figure8knot-rose-limacon-curve.svg|thumb|Symmetric depiction generated by parametric equations.]] || [[File:Superfície - bordo nó figura-oito.jpg|thumb|Mathematical surface Illustrating Figure-eight knot]]
|}

The figure-eight knot has genus 1 and is fibered. 
Therefore its complement fibers over the circle, the fibers being [[Seifert surface]]s which are 2-dimensional tori with one boundary component. 
The [[fibered knot|monodromy map]] is then a homeomorphism of the 2-torus, which can be represented in this case by the matrix <math>(\begin{smallmatrix}2&1\\1&1\end{smallmatrix})</math>.

==Invariants==
The [[Alexander polynomial]] of the figure-eight knot is
:<math>\Delta(t) = -t + 3 - t^{-1},\ </math>
the [[Alexander polynomial#Alexander–Conway polynomial|Conway polynomial]] is
:<math>\nabla(z) = 1-z^2,\ </math><ref>{{Knot Atlas|4_1}}</ref>
and the [[Jones polynomial]] is
:<math>V(q) = q^2 - q + 1 - q^{-1} + q^{-2}.\ </math>
The symmetry between <math>q</math> and <math>q^{-1}</math> in the Jones polynomial reflects the fact that the figure-eight knot is achiral.

[[File:Figure-eight knot.webm|thumb|400px|Figure-eight knot]]
==Notes==
{{Reflist|group=note}}

==References==
{{reflist}}

==Further reading==
* [[Ian Agol]], ''Bounds on exceptional Dehn filling'', [[Geometry & Topology]] 4 (2000), 431–449.  {{MathSciNet|id=1799796}}
* Chun Cao and Robert Meyerhoff, ''The orientable cusped hyperbolic 3-manifolds of minimum volume'', Inventiones Mathematicae, 146  (2001), no. 3, 451–478.   {{MathSciNet|id=1869847}}
* [[Marc Lackenby]], ''Word hyperbolic Dehn surgery'', [[Inventiones Mathematicae]] 140 (2000), no. 2, 243–282. {{MathSciNet|id=1756996}}
* [[Marc Lackenby]] and Robert Meyerhoff, [http://arxiv.org/abs/0808.1176 ''The maximal number of exceptional Dehn surgeries''], arXiv:0808.1176
* [[Robion Kirby]], [http://math.berkeley.edu/~kirby/problems.ps.gz ''Problems in low-dimensional topology''], (see problem 1.77, due to [[Cameron Gordon (mathematician)|Cameron Gordon]], for exceptional slopes)
* William Thurston, [http://msri.org/publications/books/gt3m/ ''The Geometry and Topology of Three-Manifolds''], Princeton University lecture notes (1978–1981).

==External links==
* {{Knot Atlas|4_1|date=7 May 2013}}
* {{MathWorld|title=Figure Eight Knot|urlname=FigureEightKnot}}

{{Knot theory|state=collapsed}}

[[Category:3-manifolds]]
[[Category:Double torus knots and links]]