8 1

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7 7.gif

7_7

8 2.gif

8_2

Contents

8 1.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X5,16,6,1 X7,14,8,15 X13,8,14,9 X15,6,16,7
Gauss code -1, 4, -3, 1, -5, 8, -6, 7, -2, 3, -4, 2, -7, 6, -8, 5
Dowker-Thistlethwaite code 4 10 16 14 12 2 8 6
Conway Notation [62]


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.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.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 1 ML.gif 8 1 AP.gif
[{10, 7}, {6, 8}, {7, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {9, 2}, {8, 10}, {1, 9}]

[edit Notes on presentations of 8 1]

Knot 8_1.
A graph, knot 8_1.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 1
Bridge index 2
Super bridge index \{4,5\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-7][-3]
Hyperbolic Volume 3.42721
A-Polynomial See Data:8 1/A-polynomial

[edit Notes for 8 1'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 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t+7-3 t^{-1}
Conway polynomial 1-3 z^2
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 13, 0 }
Jones polynomial q^2-q+2-2 q^{-1} +2 q^{-2} -2 q^{-3} + q^{-4} - q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) a^6-z^2 a^4-a^4-z^2 a^2-z^2+ a^{-2}
Kauffman polynomial (db, data sources) a^5 z^7+a^3 z^7+a^6 z^6+2 a^4 z^6+a^2 z^6-5 a^5 z^5-4 a^3 z^5+a z^5-5 a^6 z^4-8 a^4 z^4-2 a^2 z^4+z^4+7 a^5 z^3+5 a^3 z^3-a z^3+z^3 a^{-1} +6 a^6 z^2+7 a^4 z^2+z^2 a^{-2} -3 a^5 z-3 a^3 z-a^6-a^4- a^{-2}
The A2 invariant q^{20}+q^{18}-q^{12}-q^{10}+ q^{-2} + q^{-6} + q^{-8}
The G2 invariant q^{94}+q^{90}-q^{88}+2 q^{80}-2 q^{78}+q^{76}+q^{74}+q^{70}-q^{68}+q^{64}+q^{54}-q^{52}-q^{46}-q^{42}+q^{40}-q^{38}+q^{36}-q^{34}-2 q^{32}+q^{30}-q^{28}-q^{22}+q^{18}-q^{12}+q^8+ q^{-2} - q^{-6} + q^{-10} + q^{-14} + q^{-20} + q^{-24} + q^{-28} + q^{-34} + q^{-38}