10 6

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10 5.gif

10_5

10 7.gif

10_7

Contents

10 6.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,16,6,17 X7,18,8,19 X9,20,10,1 X17,6,18,7 X19,8,20,9 X15,10,16,11
Gauss code -1, 4, -3, 1, -5, 8, -6, 9, -7, 10, -2, 3, -4, 2, -10, 5, -8, 6, -9, 7
Dowker-Thistlethwaite code 4 12 16 18 20 14 2 10 6 8
Conway Notation [532]


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

Length is 11, width is 4,

Braid index is 4

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

[edit Notes on presentations of 10 6]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-13][1]
Hyperbolic Volume 8.39094
A-Polynomial See Data:10 6/A-polynomial

[edit Notes for 10 6's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 2
Topological 4 genus 2
Concordance genus 3
Rasmussen s-Invariant -4

[edit Notes for 10 6's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+6 t^2-7 t+7-7 t^{-1} +6 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-6 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 37, -4 }
Jones polynomial 1- q^{-1} +3 q^{-2} -4 q^{-3} +5 q^{-4} -6 q^{-5} +6 q^{-6} -5 q^{-7} +3 q^{-8} -2 q^{-9} + q^{-10}
HOMFLY-PT polynomial (db, data sources) z^4 a^8+3 z^2 a^8+a^8-z^6 a^6-4 z^4 a^6-4 z^2 a^6-a^6-z^6 a^4-4 z^4 a^4-4 z^2 a^4-2 a^4+z^4 a^2+4 z^2 a^2+3 a^2
Kauffman polynomial (db, data sources) z^4 a^{12}-2 z^2 a^{12}+2 z^5 a^{11}-4 z^3 a^{11}+z a^{11}+2 z^6 a^{10}-3 z^4 a^{10}+z^2 a^{10}+2 z^7 a^9-4 z^5 a^9+4 z^3 a^9+2 z^8 a^8-7 z^6 a^8+12 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-3 z^7 a^7+5 z^5 a^7-2 z^3 a^7+3 z^8 a^6-12 z^6 a^6+18 z^4 a^6-10 z^2 a^6+a^6+z^9 a^5-4 z^7 a^5+8 z^5 a^5-10 z^3 a^5+3 z a^5+z^8 a^4-2 z^6 a^4-3 z^4 a^4+5 z^2 a^4-2 a^4+z^7 a^3-3 z^5 a^3+2 z a^3+z^6 a^2-5 z^4 a^2+7 z^2 a^2-3 a^2
The A2 invariant q^{30}-q^{22}+q^{20}-q^{18}-q^{14}-2 q^{12}+q^{10}+2 q^6+q^4+q^2+1
The G2 invariant q^{162}-q^{160}+2 q^{158}-3 q^{156}+q^{154}-3 q^{150}+5 q^{148}-6 q^{146}+6 q^{144}-4 q^{142}+4 q^{138}-7 q^{136}+9 q^{134}-8 q^{132}+6 q^{130}-4 q^{128}+7 q^{124}-9 q^{122}+13 q^{120}-10 q^{118}+6 q^{116}-7 q^{112}+9 q^{110}-9 q^{108}+5 q^{106}+5 q^{104}-9 q^{102}+7 q^{100}-q^{98}-7 q^{96}+14 q^{94}-17 q^{92}+11 q^{90}-3 q^{88}-7 q^{86}+18 q^{84}-20 q^{82}+18 q^{80}-11 q^{78}-2 q^{76}+9 q^{74}-15 q^{72}+15 q^{70}-14 q^{68}+5 q^{66}+5 q^{64}-11 q^{62}+10 q^{60}-7 q^{58}-4 q^{56}+10 q^{54}-13 q^{52}+5 q^{50}-9 q^{46}+18 q^{44}-17 q^{42}+10 q^{40}-q^{38}-8 q^{36}+14 q^{34}-13 q^{32}+11 q^{30}-4 q^{28}+2 q^{26}+4 q^{24}-5 q^{22}+7 q^{20}-4 q^{18}+4 q^{16}+2 q^{10}-q^8+2 q^6+q^2