8 16

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8 15.gif

8_15

8 17.gif

8_17

Contents

8 16.gif
(KnotPlot image)

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Square depiction.

Knot presentations

Planar diagram presentation X6271 X14,6,15,5 X16,11,1,12 X12,7,13,8 X8394 X4,9,5,10 X10,15,11,16 X2,14,3,13
Gauss code 1, -8, 5, -6, 2, -1, 4, -5, 6, -7, 3, -4, 8, -2, 7, -3
Dowker-Thistlethwaite code 6 8 14 12 4 16 2 10
Conway Notation [.2.20]


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

Length is 8, width is 3,

Braid index is 3

8 16 ML.gif 8 16 AP.gif
[{3, 10}, {2, 6}, {8, 11}, {9, 7}, {4, 8}, {6, 9}, {5, 3}, {10, 4}, {1, 5}, {11, 2}, {7, 1}]

[edit Notes on presentations of 8 16]

Knot 8_16.
A graph, knot 8_16.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index 4
Nakanishi index 1
Maximal Thurston-Bennequin number [-8][-2]
Hyperbolic Volume 10.579
A-Polynomial See Data:8 16/A-polynomial

[edit Notes for 8 16'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 8 16'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) z^3 a^7+3 z^4 a^6-z^2 a^6+5 z^5 a^5-5 z^3 a^5+2 z a^5+5 z^6 a^4-7 z^4 a^4+4 z^2 a^4-a^4+2 z^7 a^3+3 z^5 a^3-10 z^3 a^3+4 z a^3+8 z^6 a^2-18 z^4 a^2+10 z^2 a^2-2 a^2+2 z^7 a-z^5 a-6 z^3 a+3 z a+3 z^6-8 z^4+5 z^2+z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1}
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}-2 q^{98}+3 q^{96}-4 q^{94}+2 q^{92}-q^{90}-2 q^{88}+9 q^{86}-12 q^{84}+15 q^{82}-14 q^{80}+7 q^{78}+2 q^{76}-16 q^{74}+28 q^{72}-31 q^{70}+24 q^{68}-10 q^{66}-11 q^{64}+26 q^{62}-30 q^{60}+21 q^{58}-5 q^{56}-15 q^{54}+23 q^{52}-19 q^{50}+2 q^{48}+22 q^{46}-36 q^{44}+36 q^{42}-20 q^{40}-4 q^{38}+30 q^{36}-45 q^{34}+46 q^{32}-33 q^{30}+12 q^{28}+14 q^{26}-32 q^{24}+39 q^{22}-30 q^{20}+14 q^{18}+5 q^{16}-20 q^{14}+24 q^{12}-14 q^{10}-q^8+21 q^6-29 q^4+26 q^2-6-17 q^{-2} +35 q^{-4} -37 q^{-6} +28 q^{-8} -10 q^{-10} -12 q^{-12} +24 q^{-14} -25 q^{-16} +20 q^{-18} -9 q^{-20} -2 q^{-22} +6 q^{-24} -8 q^{-26} +5 q^{-28} -2 q^{-30} + q^{-32}