9 9

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9 8.gif

9_8

9 10.gif

9_10

Contents

9 9.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X1627 X3,12,4,13 X7,16,8,17 X9,18,10,1 X17,8,18,9 X15,10,16,11 X5,14,6,15 X11,2,12,3 X13,4,14,5
Gauss code -1, 8, -2, 9, -7, 1, -3, 5, -4, 6, -8, 2, -9, 7, -6, 3, -5, 4
Dowker-Thistlethwaite code 6 12 14 16 18 2 4 10 8
Conway Notation [423]


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

Length is 10, width is 3,

Braid index is 3

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

[edit Notes on presentations of 9 9]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 2
Super bridge index \{4,6\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-16][5]
Hyperbolic Volume 8.01682
A-Polynomial See Data:9 9/A-polynomial

[edit Notes for 9 9's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 3
Topological 4 genus 3
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for 9 9's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-4 t^2+6 t-7+6 t^{-1} -4 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+8 z^4+8 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 31, -6 }
Jones polynomial  q^{-3} - q^{-4} +3 q^{-5} -4 q^{-6} +5 q^{-7} -5 q^{-8} +5 q^{-9} -4 q^{-10} +2 q^{-11} - q^{-12}
HOMFLY-PT polynomial (db, data sources) -z^4 a^{10}-3 z^2 a^{10}-2 a^{10}+z^6 a^8+4 z^4 a^8+4 z^2 a^8+a^8+z^6 a^6+5 z^4 a^6+7 z^2 a^6+2 a^6
Kauffman polynomial (db, data sources) z^3 a^{15}-z a^{15}+2 z^4 a^{14}-z^2 a^{14}+3 z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+3 z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-2 z^5 a^{11}+z^8 a^{10}-z^6 a^{10}+2 z^4 a^{10}-6 z^2 a^{10}+2 a^{10}+3 z^7 a^9-8 z^5 a^9+5 z^3 a^9-2 z a^9+z^8 a^8-3 z^6 a^8+3 z^4 a^8-3 z^2 a^8+a^8+z^7 a^7-3 z^5 a^7+z^3 a^7+z a^7+z^6 a^6-5 z^4 a^6+7 z^2 a^6-2 a^6
The A2 invariant -q^{36}-q^{32}-q^{30}-q^{26}+2 q^{24}+q^{20}+q^{18}+2 q^{14}+q^{10}
The G2 invariant q^{196}-q^{194}+2 q^{192}-2 q^{190}+q^{188}-2 q^{184}+5 q^{182}-6 q^{180}+6 q^{178}-6 q^{176}+2 q^{174}+3 q^{172}-7 q^{170}+12 q^{168}-11 q^{166}+9 q^{164}-5 q^{162}-2 q^{160}+6 q^{158}-11 q^{156}+11 q^{154}-7 q^{152}+3 q^{148}-7 q^{146}+5 q^{144}-q^{142}-8 q^{140}+9 q^{138}-12 q^{136}+5 q^{134}+5 q^{132}-15 q^{130}+20 q^{128}-18 q^{126}+10 q^{124}+q^{122}-12 q^{120}+18 q^{118}-18 q^{116}+14 q^{114}-4 q^{112}-4 q^{110}+11 q^{108}-10 q^{106}+7 q^{104}-2 q^{102}-5 q^{100}+8 q^{98}-8 q^{96}+2 q^{94}+6 q^{92}-11 q^{90}+16 q^{88}-11 q^{86}+q^{84}+7 q^{82}-11 q^{80}+16 q^{78}-12 q^{76}+6 q^{74}+2 q^{72}-5 q^{70}+10 q^{68}-7 q^{66}+5 q^{64}+2 q^{58}-2 q^{56}+2 q^{54}+q^{50}