8 3

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8 2.gif

8_2

8 4.gif

8_4

Contents

8 3.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X6271 X14,10,15,9 X10,5,11,6 X12,3,13,4 X4,11,5,12 X2,13,3,14 X16,8,1,7 X8,16,9,15
Gauss code 1, -6, 4, -5, 3, -1, 7, -8, 2, -3, 5, -4, 6, -2, 8, -7
Dowker-Thistlethwaite code 6 12 10 16 14 4 2 8
Conway Notation [44]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.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 10, width is 5,

Braid index is 5

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

[edit Notes on presentations of 8 3]

Knot 8_3.
A graph, knot 8_3.

Three dimensional invariants

Symmetry type Fully amphicheiral
Unknotting number 2
3-genus 1
Bridge index 2
Super bridge index \{4,6\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-5][-5]
Hyperbolic Volume 5.23868
A-Polynomial See Data:8 3/A-polynomial

[edit Notes for 8 3's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 8 3's four dimensional invariants]

Polynomial invariants

Alexander polynomial -4 t+9-4 t^{-1}
Conway polynomial 1-4 z^2
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 17, 0 }
Jones polynomial q^4-q^3+2 q^2-3 q+3-3 q^{-1} +2 q^{-2} - q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) a^4-z^2 a^2-2 z^2-1-z^2 a^{-2} + a^{-4}
Kauffman polynomial (db, data sources) a z^7+z^7 a^{-1} +a^2 z^6+z^6 a^{-2} +2 z^6+a^3 z^5-4 a z^5-4 z^5 a^{-1} +z^5 a^{-3} +a^4 z^4-2 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -6 z^4-2 a^3 z^3+8 a z^3+8 z^3 a^{-1} -2 z^3 a^{-3} -3 a^4 z^2+a^2 z^2+z^2 a^{-2} -3 z^2 a^{-4} +8 z^2-4 a z-4 z a^{-1} +a^4+ a^{-4} -1
The A2 invariant q^{14}+q^{12}+q^8-q^4-1- q^{-4} + q^{-8} + q^{-12} + q^{-14}
The G2 invariant q^{66}+q^{62}-q^{60}+q^{58}+q^{56}-q^{54}+2 q^{52}-q^{50}+2 q^{48}-q^{46}+q^{42}-2 q^{40}+4 q^{38}-3 q^{36}+q^{34}+q^{32}-2 q^{30}+3 q^{28}-2 q^{26}+q^{24}+2 q^{22}-2 q^{20}+q^{18}-2 q^{14}+4 q^{12}-4 q^{10}+q^8-3 q^4+3 q^2-5+3 q^{-2} -3 q^{-4} + q^{-8} -4 q^{-10} +4 q^{-12} -2 q^{-14} + q^{-18} -2 q^{-20} +2 q^{-22} + q^{-24} -2 q^{-26} +3 q^{-28} -2 q^{-30} + q^{-32} + q^{-34} -3 q^{-36} +4 q^{-38} -2 q^{-40} + q^{-42} - q^{-46} +2 q^{-48} - q^{-50} +2 q^{-52} - q^{-54} + q^{-56} + q^{-58} - q^{-60} + q^{-62} + q^{-66}