7 5

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7 4.gif

7_4

7 6.gif

7_6

Contents

7 5.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X1425 X3,10,4,11 X5,12,6,13 X7,14,8,1 X13,6,14,7 X11,8,12,9 X9,2,10,3
Gauss code -1, 7, -2, 1, -3, 5, -4, 6, -7, 2, -6, 3, -5, 4
Dowker-Thistlethwaite code 4 10 12 14 2 8 6
Conway Notation [322]


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

Length is 8, width is 3,

Braid index is 3

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

[edit Notes on presentations of 7 5]

Knot 7_5.
A graph, knot 7_5.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index 4
Nakanishi index 1
Maximal Thurston-Bennequin number Failed to parse (lexing error): \text{$\$$Failed}
Hyperbolic Volume 6.44354
A-Polynomial See Data:7 5/A-polynomial

[edit Notes for 7 5's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 2
Topological 4 genus 2
Concordance genus \textrm{ConcordanceGenus}(\textrm{Knot}(7,5))
Rasmussen s-Invariant -4

[edit Notes for 7 5's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^2-4 t+5-4 t^{-1} +2 t^{-2}
Conway polynomial 2 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 17, -4 }
Jones polynomial - q^{-9} +2 q^{-8} -3 q^{-7} +3 q^{-6} -3 q^{-5} +3 q^{-4} - q^{-3} + q^{-2}
HOMFLY-PT polynomial (db, data sources) a^8 \left(-z^2\right)-a^8+a^6 z^4+2 a^6 z^2+a^4 z^4+3 a^4 z^2+2 a^4
Kauffman polynomial (db, data sources) a^{11} z^3-a^{11} z+2 a^{10} z^4-2 a^{10} z^2+2 a^9 z^5-2 a^9 z^3+a^9 z+a^8 z^6+a^8 z^2-a^8+3 a^7 z^5-4 a^7 z^3+a^7 z+a^6 z^6-a^6 z^4+a^5 z^5-a^5 z^3-a^5 z+a^4 z^4-3 a^4 z^2+2 a^4
The A2 invariant -q^{28}-q^{22}-q^{18}+q^{16}+q^{14}+q^{12}+2 q^{10}+q^6
The G2 invariant q^{148}-q^{146}+2 q^{144}-2 q^{142}+q^{138}-2 q^{136}+5 q^{134}-5 q^{132}+4 q^{130}-2 q^{128}-3 q^{126}+4 q^{124}-6 q^{122}+5 q^{120}-3 q^{118}-q^{116}+3 q^{114}-3 q^{112}+q^{110}+2 q^{108}-5 q^{106}+4 q^{104}-3 q^{102}-2 q^{100}+5 q^{98}-7 q^{96}+8 q^{94}-7 q^{92}+2 q^{90}+2 q^{88}-6 q^{86}+6 q^{84}-7 q^{82}+4 q^{80}-2 q^{76}+3 q^{74}-3 q^{72}+2 q^{70}+3 q^{68}-5 q^{66}+3 q^{64}-2 q^{60}+7 q^{58}-5 q^{56}+5 q^{54}-q^{52}+4 q^{48}-4 q^{46}+5 q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+2 q^{34}+q^{30}