8 10

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8 9.gif

8_9

8 11.gif

8_11

Contents

8 10.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X1425 X3849 X9,15,10,14 X5,13,6,12 X13,7,14,6 X11,1,12,16 X15,11,16,10 X7283
Gauss code -1, 8, -2, 1, -4, 5, -8, 2, -3, 7, -6, 4, -5, 3, -7, 6
Dowker-Thistlethwaite code 4 8 12 2 14 16 6 10
Conway Notation [3,21,2]


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

Length is 8, width is 3,

Braid index is 3

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

[edit Notes on presentations of 8 10]

Knot 8_10.
A graph, knot 8_10.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index \{4,6\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-2][-8]
Hyperbolic Volume 8.65115
A-Polynomial See Data:8 10/A-polynomial

[edit Notes for 8 10's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 8 10's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-3 t^2+6 t-7+6 t^{-1} -3 t^{-2} + t^{-3}
Conway polynomial z^6+3 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 27, 2 }
Jones polynomial -q^6+2 q^5-4 q^4+5 q^3-4 q^2+5 q-3+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+9 z^2 a^{-2} -3 z^2 a^{-4} -3 z^2+6 a^{-2} -3 a^{-4} -2
Kauffman polynomial (db, data sources) z^7 a^{-1} +z^7 a^{-3} +5 z^6 a^{-2} +3 z^6 a^{-4} +2 z^6+a z^5+z^5 a^{-1} +3 z^5 a^{-3} +3 z^5 a^{-5} -13 z^4 a^{-2} -5 z^4 a^{-4} +2 z^4 a^{-6} -6 z^4-3 a z^3-8 z^3 a^{-1} -9 z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} +12 z^2 a^{-2} +6 z^2 a^{-4} -z^2 a^{-6} +5 z^2+2 a z+5 z a^{-1} +6 z a^{-3} +2 z a^{-5} -z a^{-7} -6 a^{-2} -3 a^{-4} -2
The A2 invariant -q^6-q^2+2 q^{-2} + q^{-4} +4 q^{-6} + q^{-8} + q^{-10} - q^{-12} -2 q^{-14} - q^{-18}
The G2 invariant q^{32}-q^{30}+3 q^{28}-4 q^{26}+2 q^{24}-q^{22}-5 q^{20}+9 q^{18}-12 q^{16}+10 q^{14}-6 q^{12}-5 q^{10}+12 q^8-16 q^6+14 q^4-9 q^2-2+9 q^{-2} -13 q^{-4} +9 q^{-6} - q^{-8} -4 q^{-10} +12 q^{-12} -7 q^{-14} +3 q^{-16} +7 q^{-18} -10 q^{-20} +19 q^{-22} -14 q^{-24} +9 q^{-26} +7 q^{-28} -12 q^{-30} +23 q^{-32} -19 q^{-34} +14 q^{-36} - q^{-38} -7 q^{-40} +13 q^{-42} -17 q^{-44} +11 q^{-46} - q^{-48} -7 q^{-50} +8 q^{-52} -8 q^{-54} -2 q^{-56} +8 q^{-58} -14 q^{-60} +9 q^{-62} -6 q^{-64} -3 q^{-66} +9 q^{-68} -14 q^{-70} +14 q^{-72} -8 q^{-74} +2 q^{-76} +2 q^{-78} -7 q^{-80} +6 q^{-82} -6 q^{-84} +5 q^{-86} -2 q^{-88} + q^{-92} -2 q^{-94} +2 q^{-96} - q^{-98} + q^{-100}