10 75

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10 74.gif

10_74

10 76.gif

10_76

Contents

10 75.gif
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Decorative representation by Petr Vodicka.
Symmetrical decorative form made from 45-degree lines and circular arcs.
Decorative heart knot.


Knot presentations

Planar diagram presentation X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X13,16,14,17 X7,15,8,14 X15,7,16,6 X17,20,18,1 X9,19,10,18 X19,9,20,8
Gauss code -1, 4, -3, 1, -2, 7, -6, 10, -9, 3, -4, 2, -5, 6, -7, 5, -8, 9, -10, 8
Dowker-Thistlethwaite code 4 10 12 14 18 2 16 6 20 8
Conway Notation [21,21,21+]


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

Length is 12, width is 5,

Braid index is 5

10 75 ML.gif 10 75 AP.gif
[{12, 3}, {2, 10}, {11, 4}, {3, 6}, {10, 12}, {7, 5}, {6, 8}, {4, 7}, {5, 1}, {9, 2}, {8, 11}, {1, 9}]

[edit Notes on presentations of 10 75]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 2
Maximal Thurston-Bennequin number [-5][-7]
Hyperbolic Volume 13.4307
A-Polynomial See Data:10 75/A-polynomial

[edit Notes for 10 75's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 75's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+7 t^2-19 t+27-19 t^{-1} +7 t^{-2} - t^{-3}
Conway polynomial -z^6+z^4+1
2nd Alexander ideal (db, data sources) \{3,t+1\}
Determinant and Signature { 81, 0 }
Jones polynomial q^6-3 q^5+6 q^4-10 q^3+12 q^2-13 q+14-10 q^{-1} +7 q^{-2} -4 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) -z^6+a^2 z^4+3 z^4 a^{-2} -3 z^4+a^2 z^2+6 z^2 a^{-2} -3 z^2 a^{-4} -4 z^2+3 a^{-2} -3 a^{-4} + a^{-6}
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +7 z^8 a^{-2} +3 z^8 a^{-4} +4 z^8+7 a z^7+13 z^7 a^{-1} +9 z^7 a^{-3} +3 z^7 a^{-5} +7 a^2 z^6-4 z^6 a^{-2} -3 z^6 a^{-4} +z^6 a^{-6} +7 z^6+4 a^3 z^5-5 a z^5-29 z^5 a^{-1} -29 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-8 a^2 z^4-21 z^4 a^{-2} -9 z^4 a^{-4} -3 z^4 a^{-6} -24 z^4-3 a^3 z^3-a z^3+17 z^3 a^{-1} +24 z^3 a^{-3} +9 z^3 a^{-5} +4 a^2 z^2+20 z^2 a^{-2} +12 z^2 a^{-4} +3 z^2 a^{-6} +15 z^2-a z-5 z a^{-1} -7 z a^{-3} -3 z a^{-5} -3 a^{-2} -3 a^{-4} - a^{-6}
The A2 invariant q^{12}-2 q^{10}+q^8-3 q^4+4 q^2+3 q^{-2} + q^{-4} - q^{-6} +2 q^{-8} -3 q^{-10} -2 q^{-16} + q^{-18} + q^{-20}
The G2 invariant q^{66}-3 q^{64}+6 q^{62}-10 q^{60}+9 q^{58}-6 q^{56}-2 q^{54}+19 q^{52}-32 q^{50}+50 q^{48}-55 q^{46}+39 q^{44}-9 q^{42}-40 q^{40}+87 q^{38}-126 q^{36}+134 q^{34}-107 q^{32}+37 q^{30}+62 q^{28}-144 q^{26}+193 q^{24}-183 q^{22}+114 q^{20}-13 q^{18}-96 q^{16}+154 q^{14}-150 q^{12}+86 q^{10}+31 q^8-114 q^6+133 q^4-79 q^2-29+154 q^{-2} -237 q^{-4} +223 q^{-6} -125 q^{-8} -26 q^{-10} +202 q^{-12} -306 q^{-14} +309 q^{-16} -212 q^{-18} +55 q^{-20} +106 q^{-22} -224 q^{-24} +248 q^{-26} -183 q^{-28} +72 q^{-30} +62 q^{-32} -144 q^{-34} +145 q^{-36} -70 q^{-38} -41 q^{-40} +132 q^{-42} -177 q^{-44} +133 q^{-46} -32 q^{-48} -92 q^{-50} +197 q^{-52} -228 q^{-54} +181 q^{-56} -76 q^{-58} -50 q^{-60} +136 q^{-62} -175 q^{-64} +153 q^{-66} -90 q^{-68} +16 q^{-70} +43 q^{-72} -71 q^{-74} +70 q^{-76} -46 q^{-78} +22 q^{-80} -11 q^{-84} +12 q^{-86} -10 q^{-88} +6 q^{-90} -2 q^{-92} + q^{-94}