7 7

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7 6.gif

7_6

8 1.gif

8_1

Contents

7 7.gif
(KnotPlot image)

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Ornamental knot
Mongolian ornament ; sum of two 7.7
Depiction with three loops

Knot presentations

Planar diagram presentation X1425 X5,10,6,11 X3948 X9,3,10,2 X11,14,12,1 X7,13,8,12 X13,7,14,6
Gauss code -1, 4, -3, 1, -2, 7, -6, 3, -4, 2, -5, 6, -7, 5
Dowker-Thistlethwaite code 4 8 10 12 2 14 6
Conway Notation [21112]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif

Length is 7, width is 4,

Braid index is 4

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

[edit Notes on presentations of 7 7]

Knot 7_7.
A graph, knot 7_7.

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 Failed to parse (lexing error): \text{$\$$Failed}
Hyperbolic Volume 7.64338
A-Polynomial See Data:7 7/A-polynomial

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

Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus 2
Rasmussen s-Invariant 0

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

Polynomial invariants

Alexander polynomial t^2+ t^{-2} -5 t-5 t^{-1} +9
Conway polynomial z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 21, 0 }
Jones polynomial q^4-2 q^3- q^{-3} +3 q^2+3 q^{-2} -4 q-3 q^{-1} +4
HOMFLY-PT polynomial (db, data sources)  a^{-4} -a^2 z^2-2 z^2 a^{-2} -2 a^{-2} +z^4+2 z^2+2
Kauffman polynomial (db, data sources) z^4 a^{-4} -2 z^2 a^{-4} + a^{-4} +2 z^5 a^{-3} +a^3 z^3-4 z^3 a^{-3} +2 z a^{-3} +z^6 a^{-2} +3 a^2 z^4+2 z^4 a^{-2} -3 a^2 z^2-6 z^2 a^{-2} +2 a^{-2} +3 a z^5+5 z^5 a^{-1} -3 a z^3-8 z^3 a^{-1} +a z+3 z a^{-1} +z^6+4 z^4-7 z^2+2
The A2 invariant -q^{10}+q^8+q^6+2 q^2+ q^{-2} - q^{-4} - q^{-6} - q^{-10} + q^{-12} + q^{-14}
The G2 invariant q^{52}-2 q^{50}+3 q^{48}-4 q^{46}+q^{42}-4 q^{40}+9 q^{38}-9 q^{36}+9 q^{34}-3 q^{32}-4 q^{30}+9 q^{28}-10 q^{26}+9 q^{24}-5 q^{22}-q^{20}+5 q^{18}-4 q^{16}+4 q^{14}+2 q^{12}-7 q^{10}+10 q^8-5 q^6-2 q^4+8 q^2-12+17 q^{-2} -11 q^{-4} +5 q^{-6} +3 q^{-8} -9 q^{-10} +15 q^{-12} -14 q^{-14} +6 q^{-16} - q^{-18} -4 q^{-20} +6 q^{-22} -6 q^{-24} + q^{-26} +3 q^{-28} -7 q^{-30} +5 q^{-32} -4 q^{-34} -5 q^{-36} +10 q^{-38} -11 q^{-40} +9 q^{-42} -4 q^{-44} - q^{-46} +7 q^{-48} -8 q^{-50} +9 q^{-52} -4 q^{-54} + q^{-56} + q^{-58} -3 q^{-60} +3 q^{-62} - q^{-64} + q^{-66}