9 18

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9 17.gif

9_17

9 19.gif

9_19

Contents

9 18.gif
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Knot presentations

Planar diagram presentation X1425 X3,12,4,13 X5,14,6,15 X9,18,10,1 X17,6,18,7 X7,16,8,17 X15,8,16,9 X13,10,14,11 X11,2,12,3
Gauss code -1, 9, -2, 1, -3, 5, -6, 7, -4, 8, -9, 2, -8, 3, -7, 6, -5, 4
Dowker-Thistlethwaite code 4 12 14 16 18 2 10 8 6
Conway Notation [3222]


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

Length is 11, width is 4,

Braid index is 4

9 18 ML.gif 9 18 AP.gif
[{11, 5}, {1, 9}, {8, 10}, {9, 11}, {10, 6}, {5, 7}, {4, 8}, {6, 3}, {2, 4}, {3, 1}, {7, 2}]

[edit Notes on presentations of 9 18]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index \{4,6\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-14][3]
Hyperbolic Volume 10.0577
A-Polynomial See Data:9 18/A-polynomial

[edit Notes for 9 18's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 18's four dimensional invariants]

Polynomial invariants

Alexander polynomial 4 t^2-10 t+13-10 t^{-1} +4 t^{-2}
Conway polynomial 4 z^4+6 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 41, -4 }
Jones polynomial  q^{-2} -2 q^{-3} +5 q^{-4} -6 q^{-5} +7 q^{-6} -7 q^{-7} +6 q^{-8} -4 q^{-9} +2 q^{-10} - q^{-11}
HOMFLY-PT polynomial (db, data sources) -z^2 a^{10}-a^{10}+z^4 a^8+z^2 a^8+2 z^4 a^6+4 z^2 a^6+a^6+z^4 a^4+2 z^2 a^4+a^4
Kauffman polynomial (db, data sources) z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+2 z^6 a^{12}-5 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-3 z^5 a^{11}+z^8 a^{10}+z^6 a^{10}-2 z^4 a^{10}-2 z^2 a^{10}+a^{10}+4 z^7 a^9-5 z^5 a^9+z^3 a^9+z^8 a^8+2 z^6 a^8-2 z^4 a^8+2 z^7 a^7+z^5 a^7-4 z^3 a^7+2 z a^7+3 z^6 a^6-4 z^4 a^6+3 z^2 a^6-a^6+2 z^5 a^5-2 z^3 a^5+z^4 a^4-2 z^2 a^4+a^4
The A2 invariant -q^{34}-2 q^{28}+q^{26}+q^{20}-q^{18}+2 q^{16}+q^{12}+2 q^{10}-q^8+q^6
The G2 invariant q^{176}-q^{174}+3 q^{172}-4 q^{170}+3 q^{168}-2 q^{166}-2 q^{164}+10 q^{162}-15 q^{160}+18 q^{158}-15 q^{156}+5 q^{154}+9 q^{152}-26 q^{150}+36 q^{148}-37 q^{146}+23 q^{144}-2 q^{142}-24 q^{140}+37 q^{138}-38 q^{136}+27 q^{134}-6 q^{132}-16 q^{130}+24 q^{128}-23 q^{126}+5 q^{124}+16 q^{122}-29 q^{120}+31 q^{118}-16 q^{116}-7 q^{114}+34 q^{112}-51 q^{110}+54 q^{108}-40 q^{106}+9 q^{104}+23 q^{102}-48 q^{100}+58 q^{98}-47 q^{96}+23 q^{94}+4 q^{92}-26 q^{90}+32 q^{88}-25 q^{86}+4 q^{84}+16 q^{82}-23 q^{80}+20 q^{78}-2 q^{76}-18 q^{74}+35 q^{72}-36 q^{70}+29 q^{68}-12 q^{66}-11 q^{64}+29 q^{62}-34 q^{60}+33 q^{58}-18 q^{56}+6 q^{54}+6 q^{52}-14 q^{50}+16 q^{48}-13 q^{46}+9 q^{44}-2 q^{42}-q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-q^{32}+q^{30}