10 137

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10 136.gif

10_136

10 138.gif

10_138

Contents

10 137.gif
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Knot presentations

Planar diagram presentation X1425 X5,10,6,11 X3948 X9,3,10,2 X16,12,17,11 X14,7,15,8 X6,15,7,16 X20,18,1,17 X18,13,19,14 X12,19,13,20
Gauss code -1, 4, -3, 1, -2, -7, 6, 3, -4, 2, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8
Dowker-Thistlethwaite code 4 8 10 -14 2 -16 -18 -6 -20 -12
Conway Notation [22,211,2-]


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

Length is 10, width is 5,

Braid index is 5

10 137 ML.gif 10 137 AP.gif
[{12, 2}, {1, 10}, {11, 6}, {10, 12}, {9, 3}, {2, 8}, {7, 9}, {8, 11}, {5, 1}, {6, 4}, {3, 5}, {4, 7}]

[edit Notes on presentations of 10 137]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-7][-3]
Hyperbolic Volume 9.25056
A-Polynomial See Data:10 137/A-polynomial

[edit Notes for 10 137'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 10 137's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^2-6 t+11-6 t^{-1} + t^{-2}
Conway polynomial z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 25, 0 }
Jones polynomial q^2-2 q+4-4 q^{-1} +4 q^{-2} -4 q^{-3} +3 q^{-4} -2 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) a^6-2 z^2 a^4-2 a^4+z^4 a^2+2 z^2 a^2+2 a^2-2 z^2-1+ a^{-2}
Kauffman polynomial (db, data sources) a^4 z^8+a^2 z^8+2 a^5 z^7+4 a^3 z^7+2 a z^7+a^6 z^6-a^4 z^6-a^2 z^6+z^6-8 a^5 z^5-15 a^3 z^5-7 a z^5-4 a^6 z^4-7 a^4 z^4-5 a^2 z^4-2 z^4+8 a^5 z^3+15 a^3 z^3+9 a z^3+2 z^3 a^{-1} +4 a^6 z^2+8 a^4 z^2+7 a^2 z^2+z^2 a^{-2} +4 z^2-3 a^5 z-5 a^3 z-3 a z-z a^{-1} -a^6-2 a^4-2 a^2- a^{-2} -1
The A2 invariant q^{20}+q^{18}-q^{16}-q^{12}-q^{10}+q^8+q^4+ q^{-2} - q^{-4} + q^{-6} + q^{-8}
The G2 invariant q^{94}-q^{92}+3 q^{90}-4 q^{88}+3 q^{86}-q^{84}-4 q^{82}+10 q^{80}-10 q^{78}+10 q^{76}-4 q^{74}-4 q^{72}+11 q^{70}-12 q^{68}+8 q^{66}-q^{64}-6 q^{62}+8 q^{60}-6 q^{58}-2 q^{56}+9 q^{54}-13 q^{52}+10 q^{50}-5 q^{48}-6 q^{46}+12 q^{44}-15 q^{42}+13 q^{40}-9 q^{38}+3 q^{36}+5 q^{34}-9 q^{32}+11 q^{30}-9 q^{28}+5 q^{26}+3 q^{24}-6 q^{22}+6 q^{20}-2 q^{18}-3 q^{16}+10 q^{14}-11 q^{12}+7 q^{10}+q^8-10 q^6+14 q^4-13 q^2+7+ q^{-2} -7 q^{-4} +7 q^{-6} -5 q^{-8} +3 q^{-10} + q^{-12} -2 q^{-14} + q^{-16} -2 q^{-20} +3 q^{-22} +2 q^{-28} - q^{-30} + q^{-32} + q^{-38}