8 21

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8 20.gif

8_20

9 1.gif

9_1

Contents

8 21.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X1425 X3849 X12,6,13,5 X13,16,14,1 X9,14,10,15 X15,10,16,11 X6,12,7,11 X7283
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 [21,21,2-]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif

Length is 8, width is 3,

Braid index is 3

8 21 ML.gif 8 21 AP.gif
[{9, 3}, {2, 7}, {6, 8}, {7, 9}, {4, 1}, {3, 6}, {5, 2}, {8, 4}, {1, 5}]

[edit Notes on presentations of 8 21]

Knot 8_21.
A graph: knot 8_21.
A part of a knot and a part of a graph.

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 [-9][1]
Hyperbolic Volume 6.78371
A-Polynomial See Data:8 21/A-polynomial

[edit Notes for 8 21's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus 2
Rasmussen s-Invariant -2

[edit Notes for 8 21's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^2+4 t-5+4 t^{-1} - t^{-2}
Conway polynomial 1-z^4
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 15, -2 }
Jones polynomial 2 q^{-1} -2 q^{-2} +3 q^{-3} -3 q^{-4} +2 q^{-5} -2 q^{-6} + q^{-7}
HOMFLY-PT polynomial (db, data sources) z^2 a^6+a^6-z^4 a^4-3 z^2 a^4-3 a^4+2 z^2 a^2+3 a^2
Kauffman polynomial (db, data sources) z^4 a^8-2 z^2 a^8+2 z^5 a^7-5 z^3 a^7+2 z a^7+z^6 a^6-z^4 a^6-a^6+3 z^5 a^5-6 z^3 a^5+4 z a^5+z^6 a^4-2 z^4 a^4+5 z^2 a^4-3 a^4+z^5 a^3-z^3 a^3+2 z a^3+3 z^2 a^2-3 a^2
The A2 invariant q^{22}-2 q^{14}-q^{12}-q^{10}+q^8+2 q^6+q^4+2 q^2
The G2 invariant q^{114}-q^{112}+2 q^{110}-3 q^{108}+q^{104}-4 q^{102}+5 q^{100}-4 q^{98}+3 q^{96}+q^{94}-4 q^{92}+5 q^{90}-2 q^{88}+2 q^{86}+2 q^{84}-4 q^{82}+4 q^{80}+2 q^{78}-2 q^{76}+4 q^{74}-4 q^{72}+3 q^{70}-4 q^{66}+3 q^{64}-7 q^{62}+5 q^{60}-4 q^{58}-3 q^{56}+2 q^{54}-6 q^{52}+4 q^{50}-5 q^{48}-q^{46}+2 q^{44}-3 q^{42}+2 q^{40}-3 q^{36}+7 q^{34}-2 q^{32}+3 q^{28}-3 q^{26}+7 q^{24}-2 q^{22}+2 q^{20}+q^{18}-q^{16}+4 q^{14}-q^{12}+2 q^{10}+q^8