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Figure-eight knot (mathematics)

In knot theory, a figure-eight knot (also called Listing's knot[1]) is the unique knot with a 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.

Figure-eight knot of practical knot-tying, with ends joined

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

 

for t varying over the real numbers (see 2D visual realization at bottom right).

The figure-eight knot is prime, alternating, rational with an associated value of 5/3,[2] and is 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[note 1] closed braid (namely, the closure of the 3-string braid σ1σ2−1σ1σ2−1), 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 F: R4R2, so (according to a theorem of John Milnor) the Milnor map of F is actually a fibration. Bernard Perron found the first such F for this knot, namely,

 

where

 

Mathematical properties

The figure-eight knot has played an important role historically (and continues to do so) in the theory of 3-manifolds. Sometime in the mid-to-late 1970s, William Thurston showed that the figure-eight was 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 surgeries on the figure-eight knot resulted in non-Haken, non-Seifert-fibered 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 volume,   (sequence A091518 in the OEIS), where   is the Lobachevsky function.[3] From this perspective, the figure-eight knot can be considered the simplest hyperbolic knot. The figure eight knot complement is a 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 Lackenby and Meyerhoff, whose proof relies on the geometrization conjecture and 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.

 
Simple squared depiction of figure-eight configuration.
 
Symmetric depiction generated by parametric equations.
 
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 surfaces which are 2-dimensional tori with one boundary component. The monodromy map is then a homeomorphism of the 2-torus, which can be represented in this case by the matrix  .

Invariants

The Alexander polynomial of the figure-eight knot is

 

the Conway polynomial is

 [4]

and the Jones polynomial is

 

The symmetry between   and   in the Jones polynomial reflects the fact that the figure-eight knot is achiral.

Figure-eight knot

Notes

  1. ^ A braid is called homogeneous if every generator   either occurs always with positive or always with negative sign.

References

  1. ^ "Listing knot - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2020-06-25.
  2. ^ Gruber, Hermann. . Rational Knots database. Archived from the original on 2006-02-09. Retrieved 5 May 2022.
  3. ^ William Thurston (March 2002), "7. Computation of volume" (PDF), The Geometry and Topology of Three-Manifolds, p. 165
  4. ^ "4_1", The Knot Atlas.

Further reading

  • Ian Agol, Bounds on exceptional Dehn filling, Geometry & Topology 4 (2000), 431–449. MR1799796
  • Chun Cao and Robert Meyerhoff, The orientable cusped hyperbolic 3-manifolds of minimum volume, Inventiones Mathematicae, 146 (2001), no. 3, 451–478. MR1869847
  • Marc Lackenby, Word hyperbolic Dehn surgery, Inventiones Mathematicae 140 (2000), no. 2, 243–282. MR1756996
  • Marc Lackenby and Robert Meyerhoff, The maximal number of exceptional Dehn surgeries, arXiv:0808.1176
  • Robion Kirby, Problems in low-dimensional topology, (see problem 1.77, due to Cameron Gordon, for exceptional slopes)
  • William Thurston, The Geometry and Topology of Three-Manifolds, Princeton University lecture notes (1978–1981).

External links

figure, eight, knot, mathematics, this, article, about, mathematical, concept, knot, figure, eight, knot, other, uses, figure, knot, theory, figure, eight, knot, also, called, listing, knot, unique, knot, with, crossing, number, four, this, makes, knot, with, . This article is about the mathematical concept For the knot see Figure eight knot For other uses see Figure 8 In knot theory a figure eight knot also called Listing s knot 1 is the unique knot with a 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 Figure eight knotCommon nameFigure eight knotArf invariant1Braid length4Braid no 3Bridge no 2Crosscap no 2Crossing no 4Genus1Hyperbolic volume2 02988Stick no 7Unknotting no 1Conway notation 22 A B notation41Dowker notation4 6 8 2Last Next31 51Otheralternating hyperbolic fibered prime fully amphichiral twistFigure eight knot of practical knot tying with ends joined Contents 1 Origin of name 2 Description 3 Mathematical properties 4 Invariants 5 Notes 6 References 7 Further reading 8 External linksOrigin of name EditThe 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 EditA simple parametric representation of the figure eight knot is as the set of all points x y z where x 2 cos 2 t cos 3 t y 2 cos 2 t sin 3 t z sin 4 t displaystyle begin aligned x amp left 2 cos 2t right cos 3t y amp left 2 cos 2t right sin 3t z amp sin 4t end aligned for t varying over the real numbers see 2D visual realization at bottom right The figure eight knot is prime alternating rational with an associated value of 5 3 2 and is 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 note 1 closed braid namely the closure of the 3 string braid s1s2 1s1s2 1 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 F R4 R2 so according to a theorem of John Milnor the Milnor map of F is actually a fibration Bernard Perron found the first such F for this knot namely F x y z t G x y z 2 t 2 2 z t displaystyle F x y z t G x y z 2 t 2 2zt where G x y z t z x 2 y 2 z 2 t 2 x 6 x 2 2 y 2 2 z 2 2 t 2 t x 2 y 6 x 2 2 y 2 2 z 2 2 t 2 displaystyle begin aligned G x y z t amp z x 2 y 2 z 2 t 2 x 6x 2 2y 2 2z 2 2t 2 amp tx sqrt 2 y 6x 2 2y 2 2z 2 2t 2 end aligned Mathematical properties EditThe figure eight knot has played an important role historically and continues to do so in the theory of 3 manifolds Sometime in the mid to late 1970s William Thurston showed that the figure eight was hyperbolic by decomposing its complement into two ideal hyperbolic tetrahedra Robert Riley and Troels Jorgensen 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 surgeries on the figure eight knot resulted in non Haken non Seifert fibered 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 volume 6 L p 3 2 02988 displaystyle 6 Lambda pi 3 approx 2 02988 sequence A091518 in the OEIS where L displaystyle Lambda is the Lobachevsky function 3 From this perspective the figure eight knot can be considered the simplest hyperbolic knot The figure eight knot complement is a 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 Lackenby and Meyerhoff whose proof relies on the geometrization conjecture and 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 Simple squared depiction of figure eight configuration Symmetric depiction generated by parametric equations Mathematical surface Illustrating Figure eight knotThe figure eight knot has genus 1 and is fibered Therefore its complement fibers over the circle the fibers being Seifert surfaces which are 2 dimensional tori with one boundary component The monodromy map is then a homeomorphism of the 2 torus which can be represented in this case by the matrix 2 1 1 1 displaystyle begin smallmatrix 2 amp 1 1 amp 1 end smallmatrix Invariants EditThe Alexander polynomial of the figure eight knot is D t t 3 t 1 displaystyle Delta t t 3 t 1 the Conway polynomial is z 1 z 2 displaystyle nabla z 1 z 2 4 and the Jones polynomial is V q q 2 q 1 q 1 q 2 displaystyle V q q 2 q 1 q 1 q 2 The symmetry between q displaystyle q and q 1 displaystyle q 1 in the Jones polynomial reflects the fact that the figure eight knot is achiral source source source source source source source source source source source source source source Figure eight knotNotes Edit A braid is called homogeneous if every generator s i displaystyle sigma i either occurs always with positive or always with negative sign References Edit Listing knot Encyclopedia of Mathematics encyclopediaofmath org Retrieved 2020 06 25 Gruber Hermann Rational Knots with 4 crossings Rational Knots database Archived from the original on 2006 02 09 Retrieved 5 May 2022 William Thurston March 2002 7 Computation of volume PDF The Geometry and Topology of Three Manifolds p 165 4 1 The Knot Atlas Further reading EditIan Agol Bounds on exceptional Dehn filling Geometry amp Topology 4 2000 431 449 MR1799796 Chun Cao and Robert Meyerhoff The orientable cusped hyperbolic 3 manifolds of minimum volume Inventiones Mathematicae 146 2001 no 3 451 478 MR1869847 Marc Lackenby Word hyperbolic Dehn surgery Inventiones Mathematicae 140 2000 no 2 243 282 MR1756996 Marc Lackenby and Robert Meyerhoff The maximal number of exceptional Dehn surgeries arXiv 0808 1176 Robion Kirby Problems in low dimensional topology see problem 1 77 due to Cameron Gordon for exceptional slopes William Thurston The Geometry and Topology of Three Manifolds Princeton University lecture notes 1978 1981 External links Edit 4 1 The Knot Atlas Accessed 7 May 2013 Weisstein Eric W Figure Eight Knot MathWorld Retrieved from https en wikipedia org w index php title Figure eight knot mathematics amp oldid 1130954395, wikipedia, wiki, book, books, library,

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