8 12

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8 11.gif

8_11

8 13.gif

8_13

Contents

8 12.gif
(KnotPlot image)

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In symmetric decorative form

Knot presentations

Planar diagram presentation X4251 X10,8,11,7 X8394 X2,9,3,10 X14,6,15,5 X16,11,1,12 X12,15,13,16 X6,14,7,13
Gauss code 1, -4, 3, -1, 5, -8, 2, -3, 4, -2, 6, -7, 8, -5, 7, -6
Dowker-Thistlethwaite code 4 8 14 10 2 16 6 12
Conway Notation [2222]


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

Length is 8, width is 5,

Braid index is 5

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

[edit Notes on presentations of 8 12]

Knot 8_12.
A graph, knot 8_12.

Three dimensional invariants

Symmetry type Fully amphicheiral
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index \{4,6\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-5][-5]
Hyperbolic Volume 8.93586
A-Polynomial See Data:8 12/A-polynomial

[edit Notes for 8 12's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus 2
Rasmussen s-Invariant 0

[edit Notes for 8 12's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^2-7 t+13-7 t^{-1} + t^{-2}
Conway polynomial z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 29, 0 }
Jones polynomial q^4-2 q^3+4 q^2-5 q+5-5 q^{-1} +4 q^{-2} -2 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) a^4-2 z^2 a^2-a^2+z^4+z^2+1-2 z^2 a^{-2} - a^{-2} + a^{-4}
Kauffman polynomial (db, data sources) a z^7+z^7 a^{-1} +2 a^2 z^6+2 z^6 a^{-2} +4 z^6+2 a^3 z^5+2 a z^5+2 z^5 a^{-1} +2 z^5 a^{-3} +a^4 z^4-a^2 z^4-z^4 a^{-2} +z^4 a^{-4} -4 z^4-3 a^3 z^3-3 a z^3-3 z^3 a^{-1} -3 z^3 a^{-3} -2 a^4 z^2-2 a^2 z^2-2 z^2 a^{-2} -2 z^2 a^{-4} +a^3 z+z a^{-3} +a^4+a^2+ a^{-2} + a^{-4} +1
The A2 invariant q^{14}+q^{12}-q^{10}+q^8-q^4+q^2-1+ q^{-2} - q^{-4} + q^{-8} - q^{-10} + q^{-12} + q^{-14}
The G2 invariant q^{66}-q^{64}+3 q^{62}-3 q^{60}+2 q^{58}-3 q^{54}+9 q^{52}-11 q^{50}+12 q^{48}-8 q^{46}+10 q^{42}-17 q^{40}+23 q^{38}-18 q^{36}+8 q^{34}+4 q^{32}-16 q^{30}+17 q^{28}-13 q^{26}+3 q^{24}+6 q^{22}-12 q^{20}+9 q^{18}-12 q^{14}+21 q^{12}-23 q^{10}+15 q^8-q^6-14 q^4+27 q^2-29+27 q^{-2} -14 q^{-4} - q^{-6} +15 q^{-8} -23 q^{-10} +21 q^{-12} -12 q^{-14} +9 q^{-18} -12 q^{-20} +6 q^{-22} +3 q^{-24} -13 q^{-26} +17 q^{-28} -16 q^{-30} +4 q^{-32} +8 q^{-34} -18 q^{-36} +23 q^{-38} -17 q^{-40} +10 q^{-42} -8 q^{-46} +12 q^{-48} -11 q^{-50} +9 q^{-52} -3 q^{-54} +2 q^{-58} -3 q^{-60} +3 q^{-62} - q^{-64} + q^{-66}