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| {{About|the mathematical concept|the knot|Figure-eight knot|other uses|Figure 8 (disambiguation){{!}}Figure 8}}
| | I am Oscar and I completely dig that title. North Dakota is our birth location. To collect coins is one of the issues I love most. Supervising is my occupation.<br><br>Also visit my web blog; [https://Www.tumblr.com/blog/mmattchuu tumblr.com] |
| {{Infobox knot theory
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| | name= Figure-eight knot
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| | practical name= Figure-eight knot
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| | image= Blue Figure-Eight Knot.png
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| | caption=
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| | arf invariant= 1
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| | braid length= 4
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| | braid number= 3
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| | bridge number= 2
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| | crosscap number= 2
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| | crossing number= 4
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| | hyperbolic volume= 2.02988
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| | linking number=
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| | stick number= 7
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| | unknotting number= 1
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| | conway_notation= [22]
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| | ab_notation= 4<sub>1</sub>
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| | dowker notation= 4, 6, 8, 2
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| | thistlethwaite=
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| | other=
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| | alternating= alternating
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| | class= hyperbolic
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| | fibered= fibered
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| | prime= prime
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| | slice=
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| | symmetry= fully amphichiral
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| | tricolorable=
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| | twist= twist
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| | last crossing= 3
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| | last order= 1
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| | next crossing= 5
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| | next order= 1
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| }}
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| [[File:Figure8knot-mathematical-knot-theory.svg|100px|thumb|Figure-eight knot of practical knot-tying, with ends joined.]]
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| In [[knot theory]], a '''figure-eight knot''' (also called '''Listing's knot''') is the unique knot with a [[crossing number (knot theory)|crossing number]] of four. This is the smallest possible crossing number except for the [[unknot]] and [[trefoil knot]]. The figure-eight knot is a [[prime knot]].
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| == Origin of name ==
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| 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.
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| == Description ==
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| A simple parametric representation of the figure-eight knot is as the set of all points (''x'',''y'',''z'') where
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| :<math> \begin{align}
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| x & = \left(2 + \cos{(2t)} \right) \cos{(3t)} \\
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| y & = \left(2 + \cos{(2t)} \right) \sin{(3t)} \\
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| z & = \sin{(4t)}
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| \end{align} </math>
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| for ''t'' varying over the real numbers (see 2D visual realization at bottom right).
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| The figure-eight knot is [[Prime knot|prime]], [[alternating knot|alternating]], [[rational knot|rational]] with an associated value
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| of 5/2, 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:
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| (1) It is a ''homogeneous''<ref group="note">A braid is called homogeneous if every
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| 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.
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| (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,
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| : <math>F(x, y, z, t)=G(x, y, z^2-t^2, 2zt),\,\!</math>
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| where
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| : <math>\begin{align}
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| G(x,y,z,t)=\ & (z(x^2+y^2+z^2+t^2)+x (6x^2-2y^2-2z^2-2t^2), \\
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| & \ t x \sqrt{2}+y (6x^2-2y^2-2z^2-2t^2)).
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| \end{align}</math>
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| == Mathematical properties ==
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| [[File:Figure8knot-math-square.svg|thumb|Simple squared depiction of figure-eight configuration.]]
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| 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 decomposing its complement into two [[ideal 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.
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| The figure-eight knot is also the hyperbolic knot whose complement has the smallest possible [[hyperbolic volume (knot)|volume]], 2.02988... according to the work of [[Chun Cao]] and [[Robert Meyerhoff]]. 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.
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| 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 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.
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| [[File:Figure8knot-rose-limacon-curve.svg|thumb|Symmetric depiction generated by parametric equations.]]
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| ==Invariants==
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| The [[Alexander polynomial]] of the figure-eight knot is
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| :<math>\Delta(t) = -t + 3 - t^{-1},\ </math>
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| the [[Alexander polynomial#Alexander–Conway polynomial|Conway polynomial]] is
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| :<math>\nabla(z) = 1-z^2,\ </math><ref>{{Knot Atlas|4_1}}</ref>
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| and the [[Jones polynomial]] is
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| :<math>V(q) = q^2 - q + 1 - q^{-1} + q^{-2}.\ </math>
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| 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.
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| ==Notes==
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| {{Reflist|group=note}}
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| *[[Ian Agol]], ''Bounds on exceptional Dehn filling'', [[Geometry & Topology]] 4 (2000), 431–449. {{MathSciNet|id=1799796}}
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| *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}}
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| *Marc Lackenby, ''Word hyperbolic Dehn surgery'', [[Inventiones Mathematicae]] 140 (2000), no. 2, 243–282. {{MathSciNet|id=1756996}}
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| *Marc Lackenby and Robert Meyerhoff, [http://front.math.ucdavis.edu/0808.1176 ''The maximal number of exceptional Dehn surgeries''], arXiv:0808.1176
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| *[[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)
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| *William Thurston, [http://msri.org/publications/books/gt3m/ ''The Geometry and Topology of Three-Manifolds''], Princeton University lecture notes (1978–1981).
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| ==External links==
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| *{{Knot Atlas|4_1|date=7 May 2013}}
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| *{{MathWorld|title=Figure Eight Knot|urlname=FigureEightKnot}}
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| {{Knot theory|state=collapsed}}
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| [[Category:3-manifolds]]
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I am Oscar and I completely dig that title. North Dakota is our birth location. To collect coins is one of the issues I love most. Supervising is my occupation.
Also visit my web blog; tumblr.com