8 7

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8 6.gif

8_6

8 8.gif

8_8

Contents

8 7.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X1425 X3,10,4,11 X11,1,12,16 X5,13,6,12 X7,15,8,14 X13,7,14,6 X15,9,16,8 X9,2,10,3
Gauss code -1, 8, -2, 1, -4, 6, -5, 7, -8, 2, -3, 4, -6, 5, -7, 3
Dowker-Thistlethwaite code 4 10 12 14 2 16 6 8
Conway Notation [4112]


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

Length is 8, width is 3,

Braid index is 3

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

[edit Notes on presentations of 8 7]

Knot 8_7.
A graph, knot 8_7.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 2
Super bridge index \{3,6\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-2][-8]
Hyperbolic Volume 7.0222
A-Polynomial See Data:8 7/A-polynomial

[edit Notes for 8 7's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus 3
Rasmussen s-Invariant 2

[edit Notes for 8 7's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-3 t^2+5 t-5+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 { 23, 2 }
Jones polynomial -q^6+2 q^5-3 q^4+4 q^3-4 q^2+4 q-2+2 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +5 z^4 a^{-2} -z^4 a^{-4} -z^4+8 z^2 a^{-2} -3 z^2 a^{-4} -3 z^2+4 a^{-2} -2 a^{-4} -1
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+a z^5-z^5 a^{-1} +2 z^5 a^{-5} -12 z^4 a^{-2} -3 z^4 a^{-4} +2 z^4 a^{-6} -7 z^4-3 a z^3-3 z^3 a^{-1} -2 z^3 a^{-3} -z^3 a^{-5} +z^3 a^{-7} +12 z^2 a^{-2} +4 z^2 a^{-4} -2 z^2 a^{-6} +6 z^2+a z+2 z a^{-1} +2 z a^{-3} -z a^{-7} -4 a^{-2} -2 a^{-4} -1
The A2 invariant -q^6+1+2 q^{-2} +2 q^{-6} + q^{-10} - q^{-14} - q^{-18}
The G2 invariant q^{32}-q^{30}+2 q^{28}-3 q^{26}+q^{24}-q^{22}-3 q^{20}+7 q^{18}-8 q^{16}+5 q^{14}-3 q^{12}-2 q^{10}+6 q^8-10 q^6+7 q^4-3 q^2+6 q^{-2} -6 q^{-4} +4 q^{-6} +3 q^{-8} - q^{-10} +4 q^{-12} -4 q^{-14} +2 q^{-16} +4 q^{-18} -5 q^{-20} +10 q^{-22} -7 q^{-24} +5 q^{-26} +3 q^{-28} -7 q^{-30} +10 q^{-32} -11 q^{-34} +8 q^{-36} -2 q^{-38} -3 q^{-40} +8 q^{-42} -8 q^{-44} +6 q^{-46} -3 q^{-50} +3 q^{-52} -3 q^{-54} - q^{-56} +4 q^{-58} -5 q^{-60} +5 q^{-62} -3 q^{-64} -2 q^{-66} +4 q^{-68} -7 q^{-70} +5 q^{-72} -5 q^{-74} +2 q^{-76} - q^{-78} -3 q^{-80} +4 q^{-82} -4 q^{-84} +4 q^{-86} -2 q^{-88} + q^{-90} -2 q^{-94} +2 q^{-96} - q^{-98} + q^{-100}