10 69

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10 68.gif

10_68

10 70.gif

10_70

Contents

10 69.gif
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Knot presentations

Planar diagram presentation X1425 X7,12,8,13 X3,11,4,10 X11,3,12,2 X13,17,14,16 X5,15,6,14 X15,7,16,6 X17,20,18,1 X9,19,10,18 X19,9,20,8
Gauss code -1, 4, -3, 1, -6, 7, -2, 10, -9, 3, -4, 2, -5, 6, -7, 5, -8, 9, -10, 8
Dowker-Thistlethwaite code 4 10 14 12 18 2 16 6 20 8
Conway Notation [211,21,21]


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

Length is 12, width is 5,

Braid index is 5

10 69 ML.gif 10 69 AP.gif
[{15, 3}, {4, 2}, {3, 7}, {1, 4}, {6, 13}, {8, 10}, {7, 9}, {5, 8}, {2, 6}, {14, 11}, {10, 12}, {9, 5}, {11, 1}, {13, 15}, {12, 14}]

[edit Notes on presentations of 10 69]

Knot 10_69.
A graph, knot 10_69.
A part of a knot and a part of a graph.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 2
Maximal Thurston-Bennequin number [-2][-10]
Hyperbolic Volume 14.1265
A-Polynomial See Data:10 69/A-polynomial

[edit Notes for 10 69'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 10 69's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-7 t^2+21 t-29+21 t^{-1} -7 t^{-2} + t^{-3}
Conway polynomial z^6-z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 87, 2 }
Jones polynomial -q^8+3 q^7-7 q^6+11 q^5-13 q^4+15 q^3-14 q^2+11 q-7+4 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +3 z^4 a^{-2} -3 z^4 a^{-4} -z^4+5 z^2 a^{-2} -5 z^2 a^{-4} +3 z^2 a^{-6} -z^2+2 a^{-2} -2 a^{-4} +2 a^{-6} - a^{-8}
Kauffman polynomial (db, data sources) z^9 a^{-3} +z^9 a^{-5} +4 z^8 a^{-2} +8 z^8 a^{-4} +4 z^8 a^{-6} +6 z^7 a^{-1} +14 z^7 a^{-3} +13 z^7 a^{-5} +5 z^7 a^{-7} +3 z^6 a^{-2} -4 z^6 a^{-4} +3 z^6 a^{-8} +4 z^6+a z^5-10 z^5 a^{-1} -32 z^5 a^{-3} -30 z^5 a^{-5} -8 z^5 a^{-7} +z^5 a^{-9} -17 z^4 a^{-2} -14 z^4 a^{-4} -9 z^4 a^{-6} -5 z^4 a^{-8} -7 z^4-a z^3+5 z^3 a^{-1} +22 z^3 a^{-3} +23 z^3 a^{-5} +5 z^3 a^{-7} -2 z^3 a^{-9} +11 z^2 a^{-2} +12 z^2 a^{-4} +7 z^2 a^{-6} +3 z^2 a^{-8} +3 z^2-z a^{-1} -4 z a^{-3} -6 z a^{-5} -2 z a^{-7} +z a^{-9} -2 a^{-2} -2 a^{-4} -2 a^{-6} - a^{-8}
The A2 invariant -q^6+2 q^4-q^2+4 q^{-2} -3 q^{-4} +2 q^{-6} - q^{-8} +2 q^{-12} -2 q^{-14} +3 q^{-16} - q^{-18} - q^{-20} +2 q^{-22} - q^{-24} - q^{-26}
The G2 invariant q^{32}-3 q^{30}+7 q^{28}-13 q^{26}+14 q^{24}-12 q^{22}-q^{20}+26 q^{18}-51 q^{16}+77 q^{14}-84 q^{12}+57 q^{10}-q^8-83 q^6+165 q^4-206 q^2+191-107 q^{-2} -23 q^{-4} +170 q^{-6} -269 q^{-8} +282 q^{-10} -199 q^{-12} +45 q^{-14} +111 q^{-16} -215 q^{-18} +217 q^{-20} -120 q^{-22} -17 q^{-24} +156 q^{-26} -210 q^{-28} +145 q^{-30} +6 q^{-32} -193 q^{-34} +324 q^{-36} -341 q^{-38} +228 q^{-40} -18 q^{-42} -207 q^{-44} +382 q^{-46} -429 q^{-48} +337 q^{-50} -147 q^{-52} -86 q^{-54} +260 q^{-56} -320 q^{-58} +260 q^{-60} -108 q^{-62} -54 q^{-64} +173 q^{-66} -190 q^{-68} +100 q^{-70} +42 q^{-72} -183 q^{-74} +249 q^{-76} -204 q^{-78} +69 q^{-80} +100 q^{-82} -232 q^{-84} +292 q^{-86} -249 q^{-88} +132 q^{-90} +7 q^{-92} -133 q^{-94} +191 q^{-96} -182 q^{-98} +127 q^{-100} -49 q^{-102} -15 q^{-104} +55 q^{-106} -69 q^{-108} +56 q^{-110} -35 q^{-112} +14 q^{-114} + q^{-116} -9 q^{-118} +9 q^{-120} -8 q^{-122} +5 q^{-124} -2 q^{-126} + q^{-128}