9 8

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9 7.gif

9_7

9 9.gif

9_9

Contents

9 8.gif
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Knot presentations

Planar diagram presentation X1425 X3849 X5,14,6,15 X9,1,10,18 X11,17,12,16 X15,13,16,12 X17,11,18,10 X13,6,14,7 X7283
Gauss code -1, 9, -2, 1, -3, 8, -9, 2, -4, 7, -5, 6, -8, 3, -6, 5, -7, 4
Dowker-Thistlethwaite code 4 8 14 2 18 16 6 12 10
Conway Notation [2412]


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.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.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 8 ML.gif 9 8 AP.gif
[{12, 2}, {1, 10}, {8, 11}, {10, 12}, {9, 3}, {2, 8}, {4, 9}, {3, 5}, {6, 4}, {5, 7}, {11, 6}, {7, 1}]

[edit Notes on presentations of 9 8]


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 [-8][-3]
Hyperbolic Volume 8.19235
A-Polynomial See Data:9 8/A-polynomial

[edit Notes for 9 8'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 8's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^2+8 t-11+8 t^{-1} -2 t^{-2}
Conway polynomial 1-2 z^4
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 31, -2 }
Jones polynomial q^3-2 q^2+3 q-4+5 q^{-1} -5 q^{-2} +5 q^{-3} -3 q^{-4} +2 q^{-5} - q^{-6}
HOMFLY-PT polynomial (db, data sources) -a^6+2 z^2 a^4+2 a^4-z^4 a^2-z^2 a^2-z^4-2 z^2-1+z^2 a^{-2} + a^{-2}
Kauffman polynomial (db, data sources) a^2 z^8+z^8+2 a^3 z^7+4 a z^7+2 z^7 a^{-1} +2 a^4 z^6+z^6 a^{-2} -z^6+2 a^5 z^5-3 a^3 z^5-13 a z^5-8 z^5 a^{-1} +2 a^6 z^4-4 a^2 z^4-4 z^4 a^{-2} -6 z^4+a^7 z^3+2 a^3 z^3+11 a z^3+8 z^3 a^{-1} -2 a^6 z^2-3 a^4 z^2+2 a^2 z^2+4 z^2 a^{-2} +7 z^2-a^7 z-a^5 z-a^3 z-3 a z-2 z a^{-1} +a^6+2 a^4- a^{-2} -1
The A2 invariant -q^{20}-q^{18}+q^{16}+q^{12}+2 q^{10}+q^6-q^4- q^{-2} + q^{-4} + q^{-10}
The G2 invariant q^{100}-q^{98}+2 q^{96}-2 q^{94}-3 q^{88}+4 q^{86}-5 q^{84}+4 q^{82}-4 q^{80}+3 q^{76}-5 q^{74}+6 q^{72}-7 q^{70}+5 q^{68}-4 q^{66}+3 q^{62}-5 q^{60}+9 q^{58}-6 q^{56}+5 q^{54}-q^{52}-2 q^{50}+7 q^{48}-5 q^{46}+4 q^{44}+3 q^{42}-4 q^{40}+7 q^{38}-3 q^{36}-3 q^{34}+11 q^{32}-13 q^{30}+10 q^{28}-5 q^{26}-5 q^{24}+14 q^{22}-17 q^{20}+14 q^{18}-10 q^{16}+7 q^{12}-12 q^{10}+12 q^8-9 q^6+3 q^4+3 q^2-7+7 q^{-2} -3 q^{-4} -2 q^{-6} +8 q^{-8} -10 q^{-10} +7 q^{-12} -8 q^{-16} +14 q^{-18} -15 q^{-20} +10 q^{-22} -2 q^{-24} -6 q^{-26} +11 q^{-28} -11 q^{-30} +10 q^{-32} -3 q^{-34} - q^{-36} +3 q^{-38} -4 q^{-40} +3 q^{-42} - q^{-44} + q^{-46}