9 23

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9 22.gif

9_22

9 24.gif

9_24

Contents

9 23.gif
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Symmetrical decorative knot
With crossings on 3x3 grid
Depiction with two axes of symmetry
Mongolian ornament (two crossings are unnecessary)
Mongolian ornament, sum of two 9.23
Logo of the ICMC-USP, Brazil
Other depicture with central symmetry by Alain Esculier
Other depicture by Alain Esculier
Other depicture by Alain Esculier

Knot presentations

Planar diagram presentation X1425 X3,10,4,11 X5,12,6,13 X7,16,8,17 X13,18,14,1 X17,14,18,15 X15,6,16,7 X11,8,12,9 X9,2,10,3
Gauss code -1, 9, -2, 1, -3, 7, -4, 8, -9, 2, -8, 3, -5, 6, -7, 4, -6, 5
Dowker-Thistlethwaite code 4 10 12 16 2 8 18 6 14
Conway Notation [22122]


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.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif

Length is 11, width is 4,

Braid index is 4

9 23 ML.gif 9 23 AP.gif
[{11, 4}, {3, 9}, {8, 10}, {9, 11}, {10, 5}, {4, 6}, {5, 2}, {1, 3}, {2, 7}, {6, 8}, {7, 1}]

[edit Notes on presentations of 9 23]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index \{4,7\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-14][3]
Hyperbolic Volume 10.6113
A-Polynomial See Data:9 23/A-polynomial

[edit Notes for 9 23'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 23's four dimensional invariants]

Polynomial invariants

Alexander polynomial 4 t^2-11 t+15-11 t^{-1} +4 t^{-2}
Conway polynomial 4 z^4+5 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 45, -4 }
Jones polynomial  q^{-2} -2 q^{-3} +5 q^{-4} -6 q^{-5} +8 q^{-6} -8 q^{-7} +6 q^{-8} -5 q^{-9} +3 q^{-10} - q^{-11}
HOMFLY-PT polynomial (db, data sources) -z^2 a^{10}+z^4 a^8-2 a^8+2 z^4 a^6+4 z^2 a^6+2 a^6+z^4 a^4+2 z^2 a^4+a^4
Kauffman polynomial (db, data sources) z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-7 z^4 a^{12}+3 z^2 a^{12}+3 z^7 a^{11}-5 z^5 a^{11}+z a^{11}+z^8 a^{10}+4 z^6 a^{10}-10 z^4 a^{10}+3 z^2 a^{10}+5 z^7 a^9-6 z^5 a^9-2 z^3 a^9+4 z a^9+z^8 a^8+4 z^6 a^8-8 z^4 a^8+6 z^2 a^8-2 a^8+2 z^7 a^7+2 z^5 a^7-6 z^3 a^7+4 z a^7+3 z^6 a^6-4 z^4 a^6+4 z^2 a^6-2 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}+q^{32}+q^{30}-2 q^{28}-2 q^{24}-q^{22}+q^{20}+3 q^{16}+q^{12}+2 q^{10}-q^8+q^6
The G2 invariant q^{176}-2 q^{174}+5 q^{172}-8 q^{170}+7 q^{168}-3 q^{166}-6 q^{164}+20 q^{162}-27 q^{160}+30 q^{158}-24 q^{156}+q^{154}+23 q^{152}-47 q^{150}+55 q^{148}-43 q^{146}+18 q^{144}+14 q^{142}-38 q^{140}+48 q^{138}-38 q^{136}+13 q^{134}+12 q^{132}-30 q^{130}+30 q^{128}-10 q^{126}-16 q^{124}+40 q^{122}-43 q^{120}+36 q^{118}-13 q^{116}-28 q^{114}+56 q^{112}-73 q^{110}+66 q^{108}-39 q^{106}-6 q^{104}+43 q^{102}-64 q^{100}+61 q^{98}-44 q^{96}+7 q^{94}+20 q^{92}-36 q^{90}+30 q^{88}-7 q^{86}-16 q^{84}+35 q^{82}-29 q^{80}+11 q^{78}+13 q^{76}-34 q^{74}+47 q^{72}-41 q^{70}+28 q^{68}-2 q^{66}-19 q^{64}+35 q^{62}-35 q^{60}+31 q^{58}-17 q^{56}+4 q^{54}+7 q^{52}-15 q^{50}+16 q^{48}-12 q^{46}+9 q^{44}-2 q^{42}-q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-q^{32}+q^{30}