10 136

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10 135.gif

10_135

10 137.gif

10_137

Contents

10 136.gif
(KnotPlot image)

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Knot presentations

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


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.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 4

10 136 ML.gif 10 136 AP.gif
[{11, 2}, {1, 7}, {9, 5}, {7, 11}, {8, 10}, {2, 9}, {6, 4}, {5, 8}, {3, 6}, {4, 1}, {10, 3}]

[edit Notes on presentations of 10 136] The knot 10_136 is the only knot in the Rolfsen Knot Table whose braid index is smaller than the width of its minimum braid.

The next such knot is K11n8.

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 [-3][-6]
Hyperbolic Volume 7.74627
A-Polynomial See Data:10 136/A-polynomial

[edit Notes for 10 136'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 10 136's four dimensional invariants]

Polynomial invariants

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