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

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}