8 11

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8 10.gif

8_10

8 12.gif

8_12

Contents

8 11.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 X9,16,10,1 X15,6,16,7 X7,14,8,15 X13,8,14,9
Gauss code -1, 4, -3, 1, -2, 6, -7, 8, -5, 3, -4, 2, -8, 7, -6, 5
Dowker-Thistlethwaite code 4 10 12 14 16 2 8 6
Conway Notation [3212]


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.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.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 11 ML.gif 8 11 AP.gif
[{10, 5}, {1, 8}, {9, 6}, {5, 7}, {8, 10}, {4, 9}, {6, 3}, {2, 4}, {3, 1}, {7, 2}]

[edit Notes on presentations of 8 11]

Knot 8_11.
A graph, knot 8_11.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 2
Super bridge index \{4,5\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-9][-1]
Hyperbolic Volume 8.28632
A-Polynomial See Data:8 11/A-polynomial

[edit Notes for 8 11's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 8 11's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^2+7 t-9+7 t^{-1} -2 t^{-2}
Conway polynomial -2 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 27, -2 }
Jones polynomial q-2+4 q^{-1} -4 q^{-2} +5 q^{-3} -5 q^{-4} +3 q^{-5} -2 q^{-6} + q^{-7}
HOMFLY-PT polynomial (db, data sources) z^2 a^6+a^6-z^4 a^4-2 z^2 a^4-2 a^4-z^4 a^2-z^2 a^2+a^2+z^2+1
Kauffman polynomial (db, data sources) z^4 a^8-2 z^2 a^8+2 z^5 a^7-4 z^3 a^7+2 z a^7+2 z^6 a^6-3 z^4 a^6+2 z^2 a^6-a^6+z^7 a^5+z^5 a^5-3 z^3 a^5+3 z a^5+4 z^6 a^4-7 z^4 a^4+6 z^2 a^4-2 a^4+z^7 a^3+z^5 a^3-2 z^3 a^3+z a^3+2 z^6 a^2-2 z^4 a^2-a^2+2 z^5 a-3 z^3 a+z^4-2 z^2+1
The A2 invariant q^{22}+q^{16}-2 q^{14}-q^{12}-q^{10}+2 q^6+2 q^2+ q^{-4}
The G2 invariant q^{114}-q^{112}+2 q^{110}-3 q^{108}+q^{106}-q^{104}-3 q^{102}+7 q^{100}-8 q^{98}+8 q^{96}-6 q^{94}+2 q^{92}+7 q^{90}-13 q^{88}+16 q^{86}-13 q^{84}+7 q^{82}+3 q^{80}-10 q^{78}+13 q^{76}-9 q^{74}+8 q^{72}+4 q^{70}-10 q^{68}+7 q^{66}-2 q^{64}-8 q^{62}+14 q^{60}-18 q^{58}+10 q^{56}-3 q^{54}-9 q^{52}+18 q^{50}-25 q^{48}+19 q^{46}-14 q^{44}-q^{42}+11 q^{40}-17 q^{38}+16 q^{36}-9 q^{34}+4 q^{32}+6 q^{30}-9 q^{28}+7 q^{26}-7 q^{22}+14 q^{20}-11 q^{18}+5 q^{16}+6 q^{14}-11 q^{12}+17 q^{10}-14 q^8+9 q^6-2 q^4-7 q^2+10-9 q^{-2} +8 q^{-4} -3 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18}