9 35

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9 34.gif

9_34

9 36.gif

9_36

Contents

9 35.gif
(KnotPlot image)

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9_35 is also known as the pretzel knot P(3,3,3).


Three-fold symmetric decorative knot
Another three-fold symmetric decorative form

Knot presentations

Planar diagram presentation X1829 X7,14,8,15 X5,16,6,17 X9,18,10,1 X15,6,16,7 X17,10,18,11 X13,2,14,3 X3,12,4,13 X11,4,12,5
Gauss code -1, 7, -8, 9, -3, 5, -2, 1, -4, 6, -9, 8, -7, 2, -5, 3, -6, 4
Dowker-Thistlethwaite code 8 12 16 14 18 4 2 6 10
Conway Notation [3,3,3]


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

Length is 14, width is 5,

Braid index is 5

9 35 ML.gif 9 35 AP.gif
[{8, 4}, {3, 7}, {4, 2}, {1, 3}, {9, 12}, {11, 8}, {12, 10}, {6, 9}, {7, 5}, {2, 6}, {5, 11}, {10, 1}]

[edit Notes on presentations of 9 35]


Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 1
Bridge index 3
Super bridge index \{4,6\}
Nakanishi index 2
Maximal Thurston-Bennequin number [-12][1]
Hyperbolic Volume 7.94058
A-Polynomial See Data:9 35/A-polynomial

[edit Notes for 9 35's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 35's four dimensional invariants]

Polynomial invariants

Alexander polynomial 7 t-13+7 t^{-1}
Conway polynomial 7 z^2+1
2nd Alexander ideal (db, data sources) \{3,t+1\}
Determinant and Signature { 27, -2 }
Jones polynomial  q^{-1} -2 q^{-2} +3 q^{-3} -4 q^{-4} +5 q^{-5} -3 q^{-6} +4 q^{-7} -3 q^{-8} + q^{-9} - q^{-10}
HOMFLY-PT polynomial (db, data sources) -a^{10}+z^2 a^8-a^8+3 z^2 a^6+3 a^6+2 z^2 a^4+z^2 a^2
Kauffman polynomial (db, data sources) z^7 a^{11}-6 z^5 a^{11}+12 z^3 a^{11}-8 z a^{11}+z^8 a^{10}-4 z^6 a^{10}+3 z^4 a^{10}+z^2 a^{10}+a^{10}+4 z^7 a^9-18 z^5 a^9+23 z^3 a^9-9 z a^9+z^8 a^8+z^6 a^8-15 z^4 a^8+16 z^2 a^8-a^8+3 z^7 a^7-8 z^5 a^7+3 z^3 a^7-z a^7+5 z^6 a^6-15 z^4 a^6+12 z^2 a^6-3 a^6+4 z^5 a^5-6 z^3 a^5+3 z^4 a^4-2 z^2 a^4+2 z^3 a^3+z^2 a^2
The A2 invariant -q^{32}-q^{30}-2 q^{26}-q^{24}+q^{22}+q^{20}+3 q^{18}+2 q^{16}+q^{14}-q^{10}+q^8-q^4+q^2
The G2 invariant q^{156}+3 q^{152}-3 q^{150}+2 q^{148}-q^{146}-2 q^{144}+7 q^{142}-9 q^{140}+6 q^{138}-2 q^{136}-2 q^{134}+8 q^{132}-12 q^{130}+5 q^{128}-2 q^{126}-3 q^{124}+3 q^{122}-10 q^{120}-2 q^{118}+4 q^{116}-2 q^{114}+q^{112}-8 q^{110}-2 q^{108}+6 q^{106}-6 q^{104}+5 q^{102}-11 q^{100}+6 q^{98}+8 q^{96}-3 q^{94}+8 q^{92}-10 q^{90}+12 q^{88}+4 q^{86}-5 q^{84}+7 q^{82}-5 q^{80}+5 q^{78}+7 q^{76}-3 q^{74}+2 q^{72}+q^{70}-2 q^{68}+4 q^{66}-6 q^{64}+3 q^{62}-2 q^{60}-2 q^{58}+4 q^{56}-4 q^{54}+3 q^{52}-2 q^{50}+q^{48}-q^{46}-q^{44}+2 q^{42}-3 q^{40}+3 q^{38}+q^{36}+q^{34}-q^{30}+2 q^{28}-2 q^{26}+2 q^{24}-q^{22}-q^{16}+q^{14}-q^{12}+q^{10}