10 70

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10 69.gif

10_69

10 71.gif

10_71

Contents

10 70.gif
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Knot presentations

Planar diagram presentation X1425 X7,10,8,11 X3948 X9,3,10,2 X5,16,6,17 X11,19,12,18 X13,1,14,20 X19,13,20,12 X17,15,18,14 X15,6,16,7
Gauss code -1, 4, -3, 1, -5, 10, -2, 3, -4, 2, -6, 8, -7, 9, -10, 5, -9, 6, -8, 7
Dowker-Thistlethwaite code 4 8 16 10 2 18 20 6 14 12
Conway Notation [22,3,2+]


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

Length is 10, width is 5,

Braid index is 5

10 70 ML.gif 10 70 AP.gif
[{13, 10}, {11, 9}, {10, 12}, {3, 11}, {8, 4}, {9, 7}, {5, 8}, {7, 13}, {4, 6}, {2, 5}, {1, 3}, {12, 2}, {6, 1}]

[edit Notes on presentations of 10 70]

Knot 10_70.
A graph, knot 10_70.
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 1
Maximal Thurston-Bennequin number [-3][-9]
Hyperbolic Volume 12.5109
A-Polynomial See Data:10 70/A-polynomial

[edit Notes for 10 70'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 70's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-7 t^2+16 t-19+16 t^{-1} -7 t^{-2} + t^{-3}
Conway polynomial z^6-z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 67, 2 }
Jones polynomial q^7-3 q^6+6 q^5-9 q^4+11 q^3-11 q^2+10 q-8+5 q^{-1} -2 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +3 z^4 a^{-2} -2 z^4 a^{-4} -2 z^4+a^2 z^2+4 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-6} -5 z^2+2 a^2+3 a^{-2} -2 a^{-4} + a^{-6} -3
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +5 z^8 a^{-2} +3 z^8 a^{-4} +2 z^8+2 a z^7+3 z^7 a^{-1} +6 z^7 a^{-3} +5 z^7 a^{-5} +a^2 z^6-5 z^6 a^{-2} +3 z^6 a^{-4} +5 z^6 a^{-6} -2 z^6-6 a z^5-10 z^5 a^{-1} -11 z^5 a^{-3} -4 z^5 a^{-5} +3 z^5 a^{-7} -4 a^2 z^4-8 z^4 a^{-2} -12 z^4 a^{-4} -6 z^4 a^{-6} +z^4 a^{-8} -7 z^4+5 a z^3+4 z^3 a^{-1} +2 z^3 a^{-3} -3 z^3 a^{-7} +5 a^2 z^2+9 z^2 a^{-2} +9 z^2 a^{-4} +4 z^2 a^{-6} -z^2 a^{-8} +10 z^2-a z+z a^{-3} +z a^{-5} +z a^{-7} -2 a^2-3 a^{-2} -2 a^{-4} - a^{-6} -3
The A2 invariant q^{10}+q^8+2 q^4-2 q^2-1+ q^{-2} -2 q^{-4} +3 q^{-6} - q^{-8} + q^{-10} -2 q^{-14} +2 q^{-16} - q^{-18} + q^{-22}
The G2 invariant q^{46}-q^{44}+4 q^{42}-5 q^{40}+6 q^{38}-5 q^{36}+12 q^{32}-23 q^{30}+36 q^{28}-37 q^{26}+25 q^{24}+5 q^{22}-42 q^{20}+80 q^{18}-97 q^{16}+85 q^{14}-40 q^{12}-32 q^{10}+96 q^8-132 q^6+124 q^4-70 q^2-6+70 q^{-2} -108 q^{-4} +90 q^{-6} -38 q^{-8} -31 q^{-10} +81 q^{-12} -86 q^{-14} +46 q^{-16} +27 q^{-18} -97 q^{-20} +140 q^{-22} -132 q^{-24} +74 q^{-26} +17 q^{-28} -112 q^{-30} +178 q^{-32} -181 q^{-34} +132 q^{-36} -36 q^{-38} -62 q^{-40} +129 q^{-42} -147 q^{-44} +107 q^{-46} -35 q^{-48} -39 q^{-50} +81 q^{-52} -76 q^{-54} +28 q^{-56} +37 q^{-58} -86 q^{-60} +94 q^{-62} -64 q^{-64} + q^{-66} +61 q^{-68} -107 q^{-70} +119 q^{-72} -89 q^{-74} +41 q^{-76} +16 q^{-78} -61 q^{-80} +80 q^{-82} -76 q^{-84} +57 q^{-86} -25 q^{-88} -3 q^{-90} +23 q^{-92} -32 q^{-94} +30 q^{-96} -21 q^{-98} +12 q^{-100} -2 q^{-102} -4 q^{-104} +5 q^{-106} -6 q^{-108} +4 q^{-110} -2 q^{-112} + q^{-114}