10 140

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10 139.gif

10_139

10 141.gif

10_141

Contents

10 140.gif
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10_140 is also known as the pretzel knot P(4,3,-3).


Knot presentations

Planar diagram presentation X1425 X3,10,4,11 X11,19,12,18 X14,5,15,6 X6,17,7,18 X16,7,17,8 X8,15,9,16 X13,1,14,20 X19,13,20,12 X9,2,10,3
Gauss code -1, 10, -2, 1, 4, -5, 6, -7, -10, 2, -3, 9, -8, -4, 7, -6, 5, 3, -9, 8
Dowker-Thistlethwaite code 4 10 -14 -16 2 18 20 -8 -6 12
Conway Notation [4,3,21-]


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

Length is 11, width is 4,

Braid index is 4

10 140 ML.gif 10 140 AP.gif
[{9, 2}, {1, 7}, {6, 8}, {7, 9}, {10, 13}, {8, 12}, {13, 11}, {12, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 10}, {11, 1}]

[edit Notes on presentations of 10 140]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index 2
Maximal Thurston-Bennequin number [-8][-1]
Hyperbolic Volume 5.21257
A-Polynomial See Data:10 140/A-polynomial

[edit Notes for 10 140'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 10 140's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^2-2 t+3-2 t^{-1} + t^{-2}
Conway polynomial z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \left\{2,t^2+t+1\right\}
Determinant and Signature { 9, 0 }
Jones polynomial 1- q^{-1} + q^{-2} - q^{-3} +2 q^{-4} - q^{-5} + q^{-6} - q^{-7}
HOMFLY-PT polynomial (db, data sources) -z^2 a^6-2 a^6+z^4 a^4+4 z^2 a^4+4 a^4-z^2 a^2-2 a^2+1
Kauffman polynomial (db, data sources) a^6 z^8+a^4 z^8+a^7 z^7+2 a^5 z^7+a^3 z^7-6 a^6 z^6-6 a^4 z^6-6 a^7 z^5-11 a^5 z^5-5 a^3 z^5+11 a^6 z^4+12 a^4 z^4+a^2 z^4+10 a^7 z^3+16 a^5 z^3+6 a^3 z^3-8 a^6 z^2-12 a^4 z^2-4 a^2 z^2-4 a^7 z-6 a^5 z-2 a^3 z+2 a^6+4 a^4+2 a^2+1
The A2 invariant -q^{22}-q^{20}-q^{18}+2 q^{14}+2 q^{12}+2 q^{10}-q^6-q^4+1+ q^{-2}
The G2 invariant q^{108}+q^{104}-q^{102}-q^{96}+q^{94}-q^{92}-q^{90}-q^{88}-q^{86}-q^{82}-4 q^{80}-4 q^{70}+q^{68}+3 q^{66}-q^{62}-q^{60}+3 q^{58}+5 q^{56}+2 q^{54}-q^{52}+q^{50}+3 q^{48}+4 q^{46}-2 q^{42}+q^{40}+3 q^{38}+q^{36}-q^{34}-2 q^{30}+q^{28}-3 q^{24}-q^{20}-q^{18}+q^{16}-q^{14}-q^{12}-q^8+q^6+2+ q^{-4} + q^{-6} + q^{-10}