10 59

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10 58.gif

10_58

10 60.gif

10_60

Contents

10 59.gif
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Two figure 8 knots on a loop, interlinked.

Knot presentations

Planar diagram presentation X4251 X10,6,11,5 X8394 X2,9,3,10 X14,8,15,7 X18,12,19,11 X20,15,1,16 X16,19,17,20 X12,18,13,17 X6,14,7,13
Gauss code 1, -4, 3, -1, 2, -10, 5, -3, 4, -2, 6, -9, 10, -5, 7, -8, 9, -6, 8, -7
Dowker-Thistlethwaite code 4 8 10 14 2 18 6 20 12 16
Conway Notation [22,211,2]


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

Length is 10, width is 5,

Braid index is 5

10 59 ML.gif 10 59 AP.gif
[{9, 12}, {11, 3}, {12, 10}, {7, 11}, {6, 8}, {5, 7}, {4, 2}, {3, 6}, {1, 4}, {2, 9}, {8, 1}, {10, 5}]

[edit Notes on presentations of 10 59]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-3][-9]
Hyperbolic Volume 13.3899
A-Polynomial See Data:10 59/A-polynomial

[edit Notes for 10 59'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 10 59's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-7 t^2+18 t-23+18 t^{-1} -7 t^{-2} + t^{-3}
Conway polynomial z^6-z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 75, 2 }
Jones polynomial q^7-3 q^6+6 q^5-10 q^4+12 q^3-12 q^2+12 q-9+6 q^{-1} -3 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+5 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-6} -4 z^2+a^2+4 a^{-2} -3 a^{-4} + a^{-6} -2
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +7 z^8 a^{-2} +4 z^8 a^{-4} +3 z^8+3 a z^7+8 z^7 a^{-1} +11 z^7 a^{-3} +6 z^7 a^{-5} +a^2 z^6-9 z^6 a^{-2} +z^6 a^{-4} +5 z^6 a^{-6} -4 z^6-9 a z^5-27 z^5 a^{-1} -28 z^5 a^{-3} -7 z^5 a^{-5} +3 z^5 a^{-7} -3 a^2 z^4-8 z^4 a^{-2} -11 z^4 a^{-4} -5 z^4 a^{-6} +z^4 a^{-8} -6 z^4+8 a z^3+21 z^3 a^{-1} +20 z^3 a^{-3} +4 z^3 a^{-5} -3 z^3 a^{-7} +3 a^2 z^2+11 z^2 a^{-2} +10 z^2 a^{-4} +3 z^2 a^{-6} -z^2 a^{-8} +8 z^2-2 a z-5 z a^{-1} -4 z a^{-3} +z a^{-7} -a^2-4 a^{-2} -3 a^{-4} - a^{-6} -2
The A2 invariant q^{10}-q^6+2 q^4-2 q^2+2 q^{-2} - q^{-4} +4 q^{-6} - q^{-8} + q^{-10} - q^{-12} -3 q^{-14} +2 q^{-16} - q^{-18} + q^{-22}
The G2 invariant q^{46}-2 q^{44}+6 q^{42}-10 q^{40}+12 q^{38}-10 q^{36}-q^{34}+23 q^{32}-44 q^{30}+64 q^{28}-63 q^{26}+33 q^{24}+18 q^{22}-83 q^{20}+135 q^{18}-149 q^{16}+112 q^{14}-28 q^{12}-76 q^{10}+158 q^8-183 q^6+145 q^4-55 q^2-52+123 q^{-2} -140 q^{-4} +86 q^{-6} +9 q^{-8} -96 q^{-10} +143 q^{-12} -111 q^{-14} +24 q^{-16} +88 q^{-18} -182 q^{-20} +220 q^{-22} -177 q^{-24} +68 q^{-26} +74 q^{-28} -192 q^{-30} +256 q^{-32} -225 q^{-34} +126 q^{-36} +6 q^{-38} -123 q^{-40} +179 q^{-42} -167 q^{-44} +85 q^{-46} +20 q^{-48} -97 q^{-50} +117 q^{-52} -74 q^{-54} -17 q^{-56} +101 q^{-58} -146 q^{-60} +127 q^{-62} -63 q^{-64} -31 q^{-66} +113 q^{-68} -154 q^{-70} +149 q^{-72} -93 q^{-74} +23 q^{-76} +41 q^{-78} -84 q^{-80} +92 q^{-82} -79 q^{-84} +52 q^{-86} -16 q^{-88} -10 q^{-90} +27 q^{-92} -32 q^{-94} +28 q^{-96} -19 q^{-98} +11 q^{-100} -2 q^{-102} -4 q^{-104} +5 q^{-106} -6 q^{-108} +4 q^{-110} -2 q^{-112} + q^{-114}