8 15

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8 14.gif

8_14

8 16.gif

8_16

Contents

8 15.gif
(KnotPlot image)

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Two trefoil knots along a closed loop, mutually interlinked. (See also 10 120.)

Symmetrical depiction.

Knot presentations

Planar diagram presentation X1425 X3849 X5,12,6,13 X13,16,14,1 X9,14,10,15 X15,10,16,11 X11,6,12,7 X7283
Gauss code -1, 8, -2, 1, -3, 7, -8, 2, -5, 6, -7, 3, -4, 5, -6, 4
Dowker-Thistlethwaite code 4 8 12 2 14 6 16 10
Conway Notation [21,21,2]


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

Length is 9, width is 4,

Braid index is 4

8 15 ML.gif 8 15 AP.gif
[{11, 3}, {2, 9}, {7, 10}, {9, 11}, {8, 4}, {3, 7}, {4, 1}, {5, 8}, {6, 2}, {10, 5}, {1, 6}]

[edit Notes on presentations of 8 15]

Knot 8_15.
A graph, knot 8_15.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 3
Super bridge index \{4,6\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-13][3]
Hyperbolic Volume 9.93065
A-Polynomial See Data:8 15/A-polynomial

[edit Notes for 8 15's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 8 15's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^2-8 t+11-8 t^{-1} +3 t^{-2}
Conway polynomial 3 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 33, -4 }
Jones polynomial  q^{-2} -2 q^{-3} +5 q^{-4} -5 q^{-5} +6 q^{-6} -6 q^{-7} +4 q^{-8} -3 q^{-9} + q^{-10}
HOMFLY-PT polynomial (db, data sources) a^{10}-3 z^2 a^8-4 a^8+2 z^4 a^6+5 z^2 a^6+3 a^6+z^4 a^4+2 z^2 a^4+a^4
Kauffman polynomial (db, data sources) z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-5 z^3 a^{11}+2 z a^{11}+3 z^6 a^{10}-3 z^4 a^{10}-a^{10}+z^7 a^9+6 z^5 a^9-14 z^3 a^9+8 z a^9+6 z^6 a^8-10 z^4 a^8+8 z^2 a^8-4 a^8+z^7 a^7+5 z^5 a^7-11 z^3 a^7+6 z a^7+3 z^6 a^6-5 z^4 a^6+5 z^2 a^6-3 a^6+2 z^5 a^5-2 z^3 a^5+z^4 a^4-2 z^2 a^4+a^4
The A2 invariant q^{32}+q^{30}-2 q^{28}-q^{26}-2 q^{24}-2 q^{22}+q^{20}+3 q^{16}+q^{14}+q^{12}+2 q^{10}-q^8+q^6
The G2 invariant q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+3 q^{154}-q^{152}-6 q^{150}+14 q^{148}-18 q^{146}+20 q^{144}-12 q^{142}-q^{140}+17 q^{138}-27 q^{136}+34 q^{134}-24 q^{132}+7 q^{130}+10 q^{128}-21 q^{126}+25 q^{124}-16 q^{122}+q^{120}+14 q^{118}-20 q^{116}+12 q^{114}-24 q^{110}+34 q^{108}-36 q^{106}+18 q^{104}-2 q^{102}-24 q^{100}+40 q^{98}-47 q^{96}+34 q^{94}-18 q^{92}-7 q^{90}+25 q^{88}-33 q^{86}+26 q^{84}-9 q^{82}-2 q^{80}+16 q^{78}-17 q^{76}+9 q^{74}+10 q^{72}-21 q^{70}+29 q^{68}-21 q^{66}+6 q^{64}+17 q^{62}-27 q^{60}+33 q^{58}-23 q^{56}+12 q^{54}+2 q^{52}-14 q^{50}+17 q^{48}-14 q^{46}+10 q^{44}-2 q^{42}-q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-q^{32}+q^{30}