7 6

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7 5.gif

7_5

7 7.gif

7_7

Contents

7 6.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X1425 X3849 X5,12,6,13 X9,1,10,14 X13,11,14,10 X11,6,12,7 X7283
Gauss code -1, 7, -2, 1, -3, 6, -7, 2, -4, 5, -6, 3, -5, 4
Dowker-Thistlethwaite code 4 8 12 2 14 6 10
Conway Notation [2212]


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

Length is 7, width is 4,

Braid index is 4

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

[edit Notes on presentations of 7 6]

Knot 7_6.
A graph, knot 7_6.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 2
Super bridge index 4
Nakanishi index 1
Maximal Thurston-Bennequin number [-8][-1]
Hyperbolic Volume 7.08493
A-Polynomial See Data:7 6/A-polynomial

[edit Notes for 7 6's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus \textrm{ConcordanceGenus}(\textrm{Knot}(7,6))
Rasmussen s-Invariant -2

[edit Notes for 7 6's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^2- t^{-2} +5 t+5 t^{-1} -7
Conway polynomial -z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 19, -2 }
Jones polynomial q-2+3 q^{-1} -3 q^{-2} +4 q^{-3} -3 q^{-4} +2 q^{-5} - q^{-6}
HOMFLY-PT polynomial (db, data sources) -a^6+2 a^4 z^2+2 a^4-a^2 z^4-2 a^2 z^2-a^2+z^2+1
Kauffman polynomial (db, data sources) a^7 z^3-a^7 z+2 a^6 z^4-2 a^6 z^2+a^6+2 a^5 z^5-a^5 z^3+a^4 z^6+2 a^4 z^4-4 a^4 z^2+2 a^4+4 a^3 z^5-6 a^3 z^3+2 a^3 z+a^2 z^6+a^2 z^4-4 a^2 z^2+a^2+2 a z^5-4 a z^3+a z+z^4-2 z^2+1
The A2 invariant -q^{20}-q^{18}+q^{16}+q^{12}+q^{10}+q^6-q^4+q^2+ q^{-4}
The G2 invariant q^{100}-q^{98}+2 q^{96}-2 q^{94}-3 q^{88}+5 q^{86}-6 q^{84}+4 q^{82}-4 q^{80}-q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-5 q^{70}+q^{68}+2 q^{66}-5 q^{64}+4 q^{62}-2 q^{58}+6 q^{56}-5 q^{54}+2 q^{52}+6 q^{50}-9 q^{48}+11 q^{46}-9 q^{44}+5 q^{42}+2 q^{40}-6 q^{38}+10 q^{36}-9 q^{34}+8 q^{32}-3 q^{30}-3 q^{28}+5 q^{26}-5 q^{24}+3 q^{22}-3 q^{18}+5 q^{16}-3 q^{14}-q^{12}+5 q^{10}-8 q^8+8 q^6-5 q^4-q^2+5-6 q^{-2} +8 q^{-4} -4 q^{-6} +2 q^{-8} + q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18}