8 4

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8 3.gif

8_3

8 5.gif

8_5

Contents

8 4.gif
(KnotPlot image)

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Somewhat symmetric representation

Knot presentations

Planar diagram presentation X6271 X14,10,15,9 X10,3,11,4 X2,13,3,14 X12,5,13,6 X16,8,1,7 X4,11,5,12 X8,16,9,15
Gauss code 1, -4, 3, -7, 5, -1, 6, -8, 2, -3, 7, -5, 4, -2, 8, -6
Dowker-Thistlethwaite code 6 10 12 16 14 4 2 8
Conway Notation [413]


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

Length is 9, width is 4,

Braid index is 4

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

[edit Notes on presentations of 8 4]

Knot 8_4.
A graph, knot 8_4.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index \{3,5\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-7][-3]
Hyperbolic Volume 5.50049
A-Polynomial See Data:8 4/A-polynomial

[edit Notes for 8 4's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 8 4's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^2+5 t-5+5 t^{-1} -2 t^{-2}
Conway polynomial -2 z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 19, -2 }
Jones polynomial q^3-q^2+2 q-3+3 q^{-1} -3 q^{-2} +3 q^{-3} -2 q^{-4} + q^{-5}
HOMFLY-PT polynomial (db, data sources) z^2 a^4+a^4-z^4 a^2-2 z^2 a^2-z^4-3 z^2-2+z^2 a^{-2} +2 a^{-2}
Kauffman polynomial (db, data sources) a z^7+z^7 a^{-1} +2 a^2 z^6+z^6 a^{-2} +3 z^6+3 a^3 z^5-a z^5-4 z^5 a^{-1} +3 a^4 z^4-3 a^2 z^4-5 z^4 a^{-2} -11 z^4+2 a^5 z^3-5 a^3 z^3-3 a z^3+4 z^3 a^{-1} +a^6 z^2-3 a^4 z^2-a^2 z^2+7 z^2 a^{-2} +10 z^2+2 a^3 z+a z-z a^{-1} +a^4-2 a^{-2} -2
The A2 invariant q^{16}+q^{10}+q^6-q^4-q^2-1- q^{-2} + q^{-4} + q^{-6} + q^{-8} + q^{-10}
The G2 invariant q^{86}-q^{84}+q^{82}-q^{80}-q^{74}+3 q^{72}-2 q^{70}+2 q^{68}-2 q^{66}+q^{62}-2 q^{60}+3 q^{58}-2 q^{56}+q^{54}+q^{50}+2 q^{48}-q^{46}+2 q^{44}+2 q^{38}-q^{36}+q^{34}+2 q^{32}-2 q^{30}+3 q^{28}-3 q^{26}+2 q^{22}-6 q^{20}+5 q^{18}-5 q^{16}+2 q^{12}-4 q^{10}+3 q^8-4 q^6+q^4-q^2-2+ q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} - q^{-14} + q^{-16} +3 q^{-18} -4 q^{-20} +4 q^{-22} -2 q^{-24} + q^{-26} +4 q^{-28} -4 q^{-30} +4 q^{-32} - q^{-34} + q^{-36} + q^{-38} -2 q^{-40} +2 q^{-42} + q^{-46}