8 2

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8 1.gif

8_1

8 3.gif

8_3

Contents

8 2.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X7,14,8,15 X9,16,10,1 X13,6,14,7 X15,8,16,9
Gauss code -1, 4, -3, 1, -2, 7, -5, 8, -6, 3, -4, 2, -7, 5, -8, 6
Dowker-Thistlethwaite code 4 10 12 14 16 2 6 8
Conway Notation [512]


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

Length is 8, width is 3,

Braid index is 3

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

[edit Notes on presentations of 8 2]

Knot 8_2.
A graph, knot 8_2.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index \{4,5\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-11][1]
Hyperbolic Volume 4.93524
A-Polynomial See Data:8 2/A-polynomial

[edit Notes for 8 2's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 2
Topological 4 genus 2
Concordance genus 3
Rasmussen s-Invariant -4

[edit Notes for 8 2's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+3 t^2-3 t+3-3 t^{-1} +3 t^{-2} - t^{-3}
Conway polynomial -z^6-3 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 17, -4 }
Jones polynomial 1- q^{-1} +2 q^{-2} -2 q^{-3} +3 q^{-4} -3 q^{-5} +2 q^{-6} -2 q^{-7} + q^{-8}
HOMFLY-PT polynomial (db, data sources) z^4 a^6+3 z^2 a^6+a^6-z^6 a^4-5 z^4 a^4-7 z^2 a^4-3 a^4+z^4 a^2+4 z^2 a^2+3 a^2
Kauffman polynomial (db, data sources) z^2 a^{10}+2 z^3 a^9-z a^9+2 z^4 a^8-z^2 a^8+2 z^5 a^7-2 z^3 a^7-z a^7+2 z^6 a^6-5 z^4 a^6+3 z^2 a^6-a^6+z^7 a^5-2 z^5 a^5-z^3 a^5+z a^5+3 z^6 a^4-12 z^4 a^4+12 z^2 a^4-3 a^4+z^7 a^3-4 z^5 a^3+3 z^3 a^3+z a^3+z^6 a^2-5 z^4 a^2+7 z^2 a^2-3 a^2
The A2 invariant q^{24}-q^{18}-q^{16}-q^{12}+q^{10}+q^6+q^4+q^2+1
The G2 invariant q^{134}-q^{132}+q^{130}-q^{128}-q^{126}-q^{122}+2 q^{120}-2 q^{118}+q^{116}+q^{110}+q^{106}-q^{104}+q^{102}+q^{96}+2 q^{92}-q^{84}+q^{82}-2 q^{78}+q^{76}-q^{74}+q^{70}-4 q^{68}+2 q^{66}-3 q^{64}-3 q^{58}+3 q^{56}-2 q^{54}+q^{52}-q^{50}-q^{48}+q^{46}-2 q^{44}+q^{42}-q^{38}+q^{36}+q^{32}+2 q^{30}-2 q^{28}+3 q^{26}-q^{24}+q^{22}+3 q^{20}-3 q^{18}+4 q^{16}+q^{12}+q^{10}-q^8+2 q^6+q^2