9 6

From Knot Atlas
Jump to: navigation, search

9 5.gif

9_5

9 7.gif

9_7

Contents

9 6.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 6's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 6 at Knotilus!


Knot presentations

Planar diagram presentation X1425 X3,12,4,13 X5,14,6,15 X7,16,8,17 X9,18,10,1 X15,6,16,7 X17,8,18,9 X13,10,14,11 X11,2,12,3
Gauss code -1, 9, -2, 1, -3, 6, -4, 7, -5, 8, -9, 2, -8, 3, -6, 4, -7, 5
Dowker-Thistlethwaite code 4 12 14 16 18 2 10 6 8
Conway Notation [522]


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

Length is 10, width is 3,

Braid index is 3

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

[edit Notes on presentations of 9 6]


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 7.2036
A-Polynomial See Data:9 6/A-polynomial

[edit Notes for 9 6'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 6's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-4 t^2+5 t-5+5 t^{-1} -4 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+8 z^4+7 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 27, -6 }
Jones polynomial  q^{-3} - q^{-4} +3 q^{-5} -3 q^{-6} +4 q^{-7} -5 q^{-8} +4 q^{-9} -3 q^{-10} +2 q^{-11} - q^{-12}
HOMFLY-PT polynomial (db, data sources) -z^4 a^{10}-3 z^2 a^{10}-a^{10}+z^6 a^8+4 z^4 a^8+3 z^2 a^8-a^8+z^6 a^6+5 z^4 a^6+7 z^2 a^6+3 a^6
Kauffman polynomial (db, data sources) z^3 a^{15}-z a^{15}+2 z^4 a^{14}-2 z^2 a^{14}+2 z^5 a^{13}-z^3 a^{13}+2 z^6 a^{12}-2 z^4 a^{12}+z^2 a^{12}+2 z^7 a^{11}-5 z^5 a^{11}+6 z^3 a^{11}-2 z a^{11}+z^8 a^{10}-2 z^6 a^{10}+2 z^4 a^{10}-3 z^2 a^{10}+a^{10}+3 z^7 a^9-10 z^5 a^9+8 z^3 a^9-z a^9+z^8 a^8-3 z^6 a^8+z^4 a^8+z^2 a^8-a^8+z^7 a^7-3 z^5 a^7+2 z a^7+z^6 a^6-5 z^4 a^6+7 z^2 a^6-3 a^6
The A2 invariant -q^{36}-2 q^{26}-q^{22}+q^{20}+2 q^{18}+q^{16}+2 q^{14}+q^{10}
The G2 invariant q^{196}-q^{194}+2 q^{192}-2 q^{190}+q^{186}-2 q^{184}+4 q^{182}-4 q^{180}+4 q^{178}-2 q^{176}-q^{174}+3 q^{172}-4 q^{170}+4 q^{168}-5 q^{166}+3 q^{164}-3 q^{162}-q^{160}+3 q^{158}-4 q^{156}+5 q^{154}-4 q^{152}+2 q^{150}-4 q^{146}+4 q^{144}-2 q^{142}-q^{140}+6 q^{138}-5 q^{136}+2 q^{134}+3 q^{132}-6 q^{130}+9 q^{128}-10 q^{126}+3 q^{124}-5 q^{120}+10 q^{118}-12 q^{116}+6 q^{114}-3 q^{112}-2 q^{110}+3 q^{108}-9 q^{106}+6 q^{104}-4 q^{102}+4 q^{98}-6 q^{96}+4 q^{94}+3 q^{92}-5 q^{90}+7 q^{88}-6 q^{86}+2 q^{84}+5 q^{82}-7 q^{80}+11 q^{78}-7 q^{76}+5 q^{74}+2 q^{72}-4 q^{70}+7 q^{68}-5 q^{66}+5 q^{64}+2 q^{58}-q^{56}+2 q^{54}+q^{50}