10 156

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10 155.gif

10_155

10 157.gif

10_157

Contents

10 156.gif
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Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X7,14,8,15 X18,9,19,10 X6,19,7,20 X16,5,17,6 X10,17,11,18 X13,8,14,9 X20,15,1,16 X2,12,3,11
Gauss code 1, -10, 2, -1, 6, -5, -3, 8, 4, -7, 10, -2, -8, 3, 9, -6, 7, -4, 5, -9
Dowker-Thistlethwaite code 4 12 16 -14 18 2 -8 20 10 6
Conway Notation [-3:2:20]


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

Length is 11, width is 4,

Braid index is 4

10 156 ML.gif 10 156 AP.gif
[{3, 10}, {2, 4}, {1, 3}, {8, 11}, {9, 5}, {10, 7}, {4, 8}, {6, 9}, {7, 2}, {11, 6}, {5, 1}]

[edit Notes on presentations of 10 156]


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 [-8][-2]
Hyperbolic Volume 11.1634
A-Polynomial See Data:10 156/A-polynomial

[edit Notes for 10 156'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 10 156's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3}
Conway polynomial z^6+2 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 35, -2 }
Jones polynomial -q^2+3 q-4+6 q^{-1} -6 q^{-2} +6 q^{-3} -5 q^{-4} +3 q^{-5} - q^{-6}
HOMFLY-PT polynomial (db, data sources) a^2 z^6-a^4 z^4+4 a^2 z^4-z^4-2 a^4 z^2+5 a^2 z^2-2 z^2-a^4+2 a^2
Kauffman polynomial (db, data sources) a^4 z^8+a^2 z^8+a^5 z^7+4 a^3 z^7+3 a z^7-a^4 z^6+2 a^2 z^6+3 z^6-a^5 z^5-9 a^3 z^5-7 a z^5+z^5 a^{-1} +3 a^6 z^4+2 a^4 z^4-9 a^2 z^4-8 z^4+a^7 z^3+4 a^5 z^3+8 a^3 z^3+3 a z^3-2 z^3 a^{-1} -2 a^6 z^2+a^4 z^2+7 a^2 z^2+4 z^2-a^7 z-2 a^5 z-2 a^3 z-a z-a^4-2 a^2
The A2 invariant -q^{18}+q^{16}-q^{14}+q^{10}-q^8+2 q^6-q^4+2 q^2+1+ q^{-4} - q^{-6}
The G2 invariant q^{100}-q^{98}+q^{96}-2 q^{90}+2 q^{88}+2 q^{86}-6 q^{84}+11 q^{82}-16 q^{80}+10 q^{78}-3 q^{76}-13 q^{74}+29 q^{72}-35 q^{70}+27 q^{68}-8 q^{66}-16 q^{64}+35 q^{62}-36 q^{60}+23 q^{58}-20 q^{54}+28 q^{52}-21 q^{50}+2 q^{48}+21 q^{46}-34 q^{44}+32 q^{42}-18 q^{40}-5 q^{38}+27 q^{36}-42 q^{34}+42 q^{32}-31 q^{30}+11 q^{28}+15 q^{26}-35 q^{24}+44 q^{22}-33 q^{20}+15 q^{18}+10 q^{16}-28 q^{14}+31 q^{12}-16 q^{10}-3 q^8+25 q^6-32 q^4+24 q^2-1-22 q^{-2} +36 q^{-4} -35 q^{-6} +22 q^{-8} -3 q^{-10} -16 q^{-12} +25 q^{-14} -22 q^{-16} +16 q^{-18} -5 q^{-20} -3 q^{-22} +5 q^{-24} -7 q^{-26} +4 q^{-28} -2 q^{-30} + q^{-32}