8 9

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8 8.gif

8_8

8 10.gif

8_10

Contents

8 9.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X6271 X14,8,15,7 X10,3,11,4 X2,13,3,14 X12,5,13,6 X4,11,5,12 X16,10,1,9 X8,16,9,15
Gauss code 1, -4, 3, -6, 5, -1, 2, -8, 7, -3, 6, -5, 4, -2, 8, -7
Dowker-Thistlethwaite code 6 10 12 14 16 4 2 8
Conway Notation [3113]


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

Length is 8, width is 3,

Braid index is 3

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

[edit Notes on presentations of 8 9]

Knot 8_9.
A graph, knot 8_9.

Three dimensional invariants

Symmetry type Fully amphicheiral
Unknotting number 1
3-genus 3
Bridge index 2
Super bridge index \{3,6\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-5][-5]
Hyperbolic Volume 7.58818
A-Polynomial See Data:8 9/A-polynomial

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

Four dimensional invariants

Smooth 4 genus 0
Topological 4 genus 0
Concordance genus 0
Rasmussen s-Invariant 0

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

Polynomial invariants

Alexander polynomial -t^3+3 t^2-5 t+7-5 t^{-1} +3 t^{-2} - t^{-3}
Conway polynomial -z^6-3 z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 25, 0 }
Jones polynomial q^4-2 q^3+3 q^2-4 q+5-4 q^{-1} +3 q^{-2} -2 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) -z^6+a^2 z^4+z^4 a^{-2} -5 z^4+3 a^2 z^2+3 z^2 a^{-2} -8 z^2+2 a^2+2 a^{-2} -3
Kauffman polynomial (db, data sources) a z^7+z^7 a^{-1} +2 a^2 z^6+2 z^6 a^{-2} +4 z^6+2 a^3 z^5+2 z^5 a^{-3} +a^4 z^4-4 a^2 z^4-4 z^4 a^{-2} +z^4 a^{-4} -10 z^4-4 a^3 z^3-a z^3-z^3 a^{-1} -4 z^3 a^{-3} -2 a^4 z^2+4 a^2 z^2+4 z^2 a^{-2} -2 z^2 a^{-4} +12 z^2+a^3 z+a z+z a^{-1} +z a^{-3} -2 a^2-2 a^{-2} -3
The A2 invariant q^{12}+q^8-q^4+q^2-1+ q^{-2} - q^{-4} + q^{-8} + q^{-12}
The G2 invariant q^{66}-q^{64}+2 q^{62}-3 q^{60}+q^{58}-3 q^{54}+6 q^{52}-6 q^{50}+7 q^{48}-5 q^{46}+6 q^{42}-10 q^{40}+12 q^{38}-8 q^{36}+4 q^{34}+3 q^{32}-6 q^{30}+11 q^{28}-6 q^{26}+2 q^{24}+4 q^{22}-7 q^{20}+5 q^{18}-8 q^{14}+11 q^{12}-10 q^{10}+9 q^8-3 q^6-10 q^4+14 q^2-17+14 q^{-2} -10 q^{-4} -3 q^{-6} +9 q^{-8} -10 q^{-10} +11 q^{-12} -8 q^{-14} +5 q^{-18} -7 q^{-20} +4 q^{-22} +2 q^{-24} -6 q^{-26} +11 q^{-28} -6 q^{-30} +3 q^{-32} +4 q^{-34} -8 q^{-36} +12 q^{-38} -10 q^{-40} +6 q^{-42} -5 q^{-46} +7 q^{-48} -6 q^{-50} +6 q^{-52} -3 q^{-54} + q^{-58} -3 q^{-60} +2 q^{-62} - q^{-64} + q^{-66}