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8 20

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8 19.gif

8_19

8 21.gif

8_21

Contents

8 20.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 8 20's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8_20's page at Knotilus!

Visit 8 20's page at the original Knot Atlas!

8_20 is also known as the pretzel knot P(3,-3,2).

Its complement contains no complete totally geodesic immersed surfaces.[citation needed]


Knot presentations

Planar diagram presentation X4251 X8493 X5,12,6,13 X13,16,14,1 X9,14,10,15 X15,10,16,11 X11,6,12,7 X2837
Gauss code 1, -8, 2, -1, -3, 7, 8, -2, -5, 6, -7, 3, -4, 5, -6, 4
Dowker-Thistlethwaite code 4 8 -12 2 -14 -6 -16 -10
Conway Notation [3,21,2-]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif

Length is 8, width is 3,

Braid index is 3

8 20 ML.gif 8 20 AP.gif
[{3, 8}, {2, 4}, {1, 3}, {11, 9}, {8, 10}, {9, 5}, {4, 6}, {5, 7}, {6, 11}, {10, 2}, {7, 1}]

[edit Notes on presentations of 8 20]

Knot 8_20.
A graph, knot 8_20.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index 4
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-2]
Hyperbolic Volume 4.1249
A-Polynomial See Data:8 20/A-polynomial

[edit Notes for 8 20's three dimensional invariants]
8_20 ribbon diagram from A. Kawauchi's text.

Ribbon diagram for 8_20

Four dimensional invariants

Smooth 4 genus 0
Topological 4 genus 0
Concordance genus 0
Rasmussen s-Invariant 0

[edit Notes for 8 20's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^2-2 t+3-2 t^{-1} + t^{-2}
Conway polynomial z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 9, 0 }
Jones polynomial -q+2- q^{-1} +2 q^{-2} - q^{-3} + q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) -z^2 a^4-2 a^4+z^4 a^2+4 z^2 a^2+4 a^2-z^2-1
Kauffman polynomial (db, data sources) a^4 z^6+a^2 z^6+a^5 z^5+2 a^3 z^5+a z^5-4 a^4 z^4-4 a^2 z^4-4 a^5 z^3-7 a^3 z^3-3 a z^3+4 a^4 z^2+6 a^2 z^2+2 z^2+3 a^5 z+5 a^3 z+3 a z+z a^{-1} -2 a^4-4 a^2-1
The A2 invariant -q^{16}-q^{14}-q^{12}+2 q^8+2 q^6+2 q^4+q^2- q^{-4}
The G2 invariant q^{80}+q^{76}-q^{74}-2 q^{68}+q^{66}-q^{64}-q^{62}-q^{60}-2 q^{58}-3 q^{52}-q^{50}-q^{48}+q^{44}-2 q^{42}+q^{40}+q^{38}+2 q^{36}+q^{34}+2 q^{30}+2 q^{28}+3 q^{26}+2 q^{22}+3 q^{20}+q^{18}+q^{16}+q^{14}+3 q^{10}-2 q^6+q^4+1- q^{-2} -2 q^{-4} - q^{-12} - q^{-14} - q^{-20} + q^{-24}