10 131

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10 130.gif

10_130

10 132.gif

10_132

Contents

10 131.gif
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Knot presentations

Planar diagram presentation X1425 X3849 X14,6,15,5 X15,20,16,1 X9,16,10,17 X19,10,20,11 X11,18,12,19 X17,12,18,13 X6,14,7,13 X7283
Gauss code -1, 10, -2, 1, 3, -9, -10, 2, -5, 6, -7, 8, 9, -3, -4, 5, -8, 7, -6, 4
Dowker-Thistlethwaite code 4 8 -14 2 16 18 -6 20 12 10
Conway Notation [311,21,2-]


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

Length is 11, width is 4,

Braid index is 4

10 131 ML.gif 10 131 AP.gif
[{11, 6}, {5, 9}, {8, 10}, {9, 11}, {7, 1}, {6, 8}, {10, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 7}]

[edit Notes on presentations of 10 131]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-11][1]
Hyperbolic Volume 9.46502
A-Polynomial See Data:10 131/A-polynomial

[edit Notes for 10 131's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 131's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^2+8 t-11+8 t^{-1} -2 t^{-2}
Conway polynomial 1-2 z^4
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 31, -2 }
Jones polynomial 2 q^{-1} -3 q^{-2} +5 q^{-3} -5 q^{-4} +5 q^{-5} -5 q^{-6} +3 q^{-7} -2 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+a^8-z^4 a^6-2 z^2 a^6-2 a^6-z^4 a^4-z^2 a^4+2 z^2 a^2+2 a^2
Kauffman polynomial (db, data sources) z^6 a^{10}-4 z^4 a^{10}+4 z^2 a^{10}+2 z^7 a^9-8 z^5 a^9+9 z^3 a^9-3 z a^9+z^8 a^8-z^6 a^8-4 z^4 a^8+2 z^2 a^8+a^8+4 z^7 a^7-12 z^5 a^7+10 z^3 a^7-5 z a^7+z^8 a^6-2 z^4 a^6-3 z^2 a^6+2 a^6+2 z^7 a^5-3 z^5 a^5+2 z^3 a^5-z a^5+2 z^6 a^4-2 z^4 a^4+2 z^2 a^4+z^5 a^3+z^3 a^3+z a^3+3 z^2 a^2-2 a^2
The A2 invariant q^{28}+q^{22}-2 q^{20}-q^{18}-q^{16}-q^{14}+q^{12}+2 q^8+q^6+2 q^2
The G2 invariant q^{142}-q^{140}+3 q^{138}-5 q^{136}+3 q^{134}-2 q^{132}-4 q^{130}+10 q^{128}-14 q^{126}+14 q^{124}-7 q^{122}-4 q^{120}+14 q^{118}-19 q^{116}+18 q^{114}-8 q^{112}-2 q^{110}+13 q^{108}-15 q^{106}+12 q^{104}+q^{102}-8 q^{100}+14 q^{98}-11 q^{96}+2 q^{94}+7 q^{92}-15 q^{90}+19 q^{88}-18 q^{86}+9 q^{84}+2 q^{82}-17 q^{80}+21 q^{78}-25 q^{76}+14 q^{74}-4 q^{72}-11 q^{70}+16 q^{68}-18 q^{66}+11 q^{64}+q^{62}-12 q^{60}+14 q^{58}-9 q^{56}-q^{54}+11 q^{52}-15 q^{50}+14 q^{48}-4 q^{46}-3 q^{44}+10 q^{42}-14 q^{40}+14 q^{38}-7 q^{36}+2 q^{34}+4 q^{32}-7 q^{30}+8 q^{28}-5 q^{26}+6 q^{24}-q^{22}+2 q^{18}-2 q^{16}+3 q^{14}-q^{12}+2 q^{10}+q^8