10 74

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10 73.gif

10_73

10 75.gif

10_75

Contents

10 74.gif
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Knot presentations

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


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
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Length is 14, width is 5,

Braid index is 5

10 74 ML.gif 10 74 AP.gif
[{13, 2}, {1, 7}, {8, 3}, {2, 6}, {7, 13}, {9, 12}, {11, 8}, {12, 10}, {5, 9}, {6, 4}, {3, 5}, {4, 11}, {10, 1}]

[edit Notes on presentations of 10 74]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index 2
Maximal Thurston-Bennequin number [-11][-1]
Hyperbolic Volume 12.006
A-Polynomial See Data:10 74/A-polynomial

[edit Notes for 10 74'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 74's four dimensional invariants]

Polynomial invariants

Alexander polynomial -4 t^2+16 t-23+16 t^{-1} -4 t^{-2}
Conway polynomial 1-4 z^4
2nd Alexander ideal (db, data sources) \{3,t+1\}
Determinant and Signature { 63, -2 }
Jones polynomial q-3+6 q^{-1} -8 q^{-2} +11 q^{-3} -10 q^{-4} +9 q^{-5} -8 q^{-6} +4 q^{-7} -2 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+a^8-z^4 a^6-z^2 a^6-2 a^6-2 z^4 a^4-2 z^2 a^4-z^4 a^2+z^2 a^2+2 a^2+z^2
Kauffman polynomial (db, data sources) z^6 a^{10}-4 z^4 a^{10}+4 z^2 a^{10}+2 z^7 a^9-7 z^5 a^9+8 z^3 a^9-4 z a^9+2 z^8 a^8-4 z^6 a^8+z^2 a^8+a^8+z^9 a^7+2 z^7 a^7-10 z^5 a^7+11 z^3 a^7-8 z a^7+5 z^8 a^6-9 z^6 a^6+3 z^4 a^6-z^2 a^6+2 a^6+z^9 a^5+5 z^7 a^5-12 z^5 a^5+9 z^3 a^5-4 z a^5+3 z^8 a^4+z^6 a^4-9 z^4 a^4+8 z^2 a^4+5 z^7 a^3-6 z^5 a^3+3 z^3 a^3+5 z^6 a^2-7 z^4 a^2+5 z^2 a^2-2 a^2+3 z^5 a-3 z^3 a+z^4-z^2
The A2 invariant q^{28}+2 q^{22}-3 q^{20}-2 q^{18}-2 q^{14}+2 q^{12}+2 q^8+2 q^6-q^4+3 q^2-1- q^{-2} + q^{-4}
The G2 invariant q^{142}-q^{140}+3 q^{138}-5 q^{136}+4 q^{134}-5 q^{132}-q^{130}+9 q^{128}-18 q^{126}+27 q^{124}-29 q^{122}+20 q^{120}+3 q^{118}-30 q^{116}+57 q^{114}-75 q^{112}+68 q^{110}-34 q^{108}-17 q^{106}+69 q^{104}-100 q^{102}+106 q^{100}-58 q^{98}+7 q^{96}+44 q^{94}-85 q^{92}+83 q^{90}-41 q^{88}-16 q^{86}+55 q^{84}-70 q^{82}+48 q^{80}+10 q^{78}-72 q^{76}+98 q^{74}-107 q^{72}+67 q^{70}-3 q^{68}-85 q^{66}+135 q^{64}-145 q^{62}+115 q^{60}-41 q^{58}-41 q^{56}+98 q^{54}-119 q^{52}+99 q^{50}-45 q^{48}-17 q^{46}+65 q^{44}-65 q^{42}+38 q^{40}+19 q^{38}-59 q^{36}+75 q^{34}-56 q^{32}+10 q^{30}+39 q^{28}-80 q^{26}+98 q^{24}-78 q^{22}+42 q^{20}+9 q^{18}-48 q^{16}+66 q^{14}-66 q^{12}+51 q^{10}-26 q^8-q^6+19 q^4-29 q^2+28-19 q^{-2} +12 q^{-4} -2 q^{-6} -3 q^{-8} +5 q^{-10} -6 q^{-12} +4 q^{-14} -2 q^{-16} + q^{-18}