9 12

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9 11.gif

9_11

9 13.gif

9_13

Contents

9 12.gif
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Knot presentations

Planar diagram presentation X1425 X3,10,4,11 X5,16,6,17 X11,1,12,18 X17,13,18,12 X7,14,8,15 X13,8,14,9 X15,6,16,7 X9,2,10,3
Gauss code -1, 9, -2, 1, -3, 8, -6, 7, -9, 2, -4, 5, -7, 6, -8, 3, -5, 4
Dowker-Thistlethwaite code 4 10 16 14 2 18 8 6 12
Conway Notation [4212]


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

Length is 10, width is 5,

Braid index is 5

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

[edit Notes on presentations of 9 12]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 2
Super bridge index \{4,6\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-10][-1]
Hyperbolic Volume 8.83664
A-Polynomial See Data:9 12/A-polynomial

[edit Notes for 9 12's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 12's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^2+9 t-13+9 t^{-1} -2 t^{-2}
Conway polynomial -2 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 35, -2 }
Jones polynomial q-2+4 q^{-1} -5 q^{-2} +6 q^{-3} -6 q^{-4} +5 q^{-5} -3 q^{-6} +2 q^{-7} - q^{-8}
HOMFLY-PT polynomial (db, data sources) -a^8+2 z^2 a^6+2 a^6-z^4 a^4-z^2 a^4-a^4-z^4 a^2-z^2 a^2+z^2+1
Kauffman polynomial (db, data sources) z^5 a^9-3 z^3 a^9+z a^9+2 z^6 a^8-6 z^4 a^8+4 z^2 a^8-a^8+2 z^7 a^7-5 z^5 a^7+3 z^3 a^7-z a^7+z^8 a^6-5 z^4 a^6+7 z^2 a^6-2 a^6+4 z^7 a^5-11 z^5 a^5+13 z^3 a^5-4 z a^5+z^8 a^4-z^4 a^4+3 z^2 a^4-a^4+2 z^7 a^3-3 z^5 a^3+4 z^3 a^3-2 z a^3+2 z^6 a^2-z^4 a^2-2 z^2 a^2+2 z^5 a-3 z^3 a+z^4-2 z^2+1
The A2 invariant -q^{26}-q^{24}+q^{22}+q^{18}+2 q^{16}-q^{14}-q^{10}+q^6-q^4+2 q^2+ q^{-4}
The G2 invariant q^{128}-q^{126}+2 q^{124}-3 q^{122}+2 q^{120}-2 q^{118}-2 q^{116}+7 q^{114}-10 q^{112}+10 q^{110}-10 q^{108}+4 q^{106}+4 q^{104}-15 q^{102}+20 q^{100}-20 q^{98}+14 q^{96}-q^{94}-12 q^{92}+19 q^{90}-18 q^{88}+15 q^{86}-3 q^{84}-10 q^{82}+14 q^{80}-11 q^{78}+2 q^{76}+10 q^{74}-16 q^{72}+18 q^{70}-8 q^{68}-4 q^{66}+17 q^{64}-26 q^{62}+29 q^{60}-21 q^{58}+6 q^{56}+11 q^{54}-23 q^{52}+30 q^{50}-25 q^{48}+13 q^{46}-13 q^{42}+16 q^{40}-14 q^{38}+3 q^{36}+8 q^{34}-13 q^{32}+10 q^{30}-2 q^{28}-9 q^{26}+17 q^{24}-19 q^{22}+15 q^{20}-6 q^{18}-5 q^{16}+14 q^{14}-16 q^{12}+17 q^{10}-10 q^8+5 q^6+q^4-7 q^2+9-8 q^{-2} +7 q^{-4} -3 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18}