8 8

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8 7.gif

8_7

8 9.gif

8_9

Contents

8 8.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X1425 X3849 X11,15,12,14 X5,13,6,12 X13,7,14,6 X9,1,10,16 X15,11,16,10 X7283
Gauss code -1, 8, -2, 1, -4, 5, -8, 2, -6, 7, -3, 4, -5, 3, -7, 6
Dowker-Thistlethwaite code 4 8 12 2 16 14 6 10
Conway Notation [2312]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif

Length is 9, width is 4,

Braid index is 4

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

[edit Notes on presentations of 8 8]

Knot 8_8.
A graph, knot 8_8.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index \{4,5\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-4][-6]
Hyperbolic Volume 7.80134
A-Polynomial See Data:8 8/A-polynomial

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

Polynomial invariants

Alexander polynomial 2 t^2-6 t+9-6 t^{-1} +2 t^{-2}
Conway polynomial 2 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 25, 0 }
Jones polynomial -q^5+2 q^4-3 q^3+4 q^2-4 q+5-3 q^{-1} +2 q^{-2} - q^{-3}
HOMFLY-PT polynomial (db, data sources) z^4 a^{-2} +z^4-a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} +2 z^2-a^2+ a^{-2} - a^{-4} +2
Kauffman polynomial (db, data sources) z^7 a^{-1} +z^7 a^{-3} +4 z^6 a^{-2} +2 z^6 a^{-4} +2 z^6+2 a z^5+z^5 a^{-1} +z^5 a^{-5} +2 a^2 z^4-9 z^4 a^{-2} -6 z^4 a^{-4} -z^4+a^3 z^3-3 z^3 a^{-1} -5 z^3 a^{-3} -3 z^3 a^{-5} -2 a^2 z^2+5 z^2 a^{-2} +4 z^2 a^{-4} -z^2-a^3 z-a z+z a^{-1} +3 z a^{-3} +2 z a^{-5} +a^2- a^{-2} - a^{-4} +2
The A2 invariant -q^{10}-q^4+2 q^2+1+2 q^{-2} + q^{-4} + q^{-8} - q^{-10} - q^{-16}
The G2 invariant q^{52}-q^{50}+2 q^{48}-2 q^{46}-3 q^{40}+4 q^{38}-5 q^{36}+4 q^{34}-5 q^{32}+q^{30}+3 q^{28}-6 q^{26}+8 q^{24}-9 q^{22}+7 q^{20}-4 q^{18}-3 q^{16}+7 q^{14}-8 q^{12}+9 q^{10}-q^8-3 q^6+6 q^4-3 q^2+1+6 q^{-2} -10 q^{-4} +12 q^{-6} -5 q^{-8} +10 q^{-12} -14 q^{-14} +17 q^{-16} -11 q^{-18} +3 q^{-20} +4 q^{-22} -9 q^{-24} +14 q^{-26} -10 q^{-28} +6 q^{-30} + q^{-32} -5 q^{-34} +7 q^{-36} -5 q^{-38} +4 q^{-42} -8 q^{-44} +7 q^{-46} -3 q^{-48} -4 q^{-50} +10 q^{-52} -13 q^{-54} +10 q^{-56} -5 q^{-58} -4 q^{-60} +7 q^{-62} -10 q^{-64} +9 q^{-66} -5 q^{-68} +2 q^{-72} -4 q^{-74} +3 q^{-76} - q^{-78} + q^{-80}