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10 125

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10 124.gif

10_124

10 126.gif

10_126

Contents

10 125.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 125's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_125's page at Knotilus!

Visit 10 125's page at the original Knot Atlas!

10_125 is also known as the pretzel knot P(5,-3,2).


Knot presentations

Planar diagram presentation X1425 X3849 X5,14,6,15 X20,16,1,15 X16,10,17,9 X18,12,19,11 X10,18,11,17 X12,20,13,19 X13,6,14,7 X7283
Gauss code -1, 10, -2, 1, -3, 9, -10, 2, 5, -7, 6, -8, -9, 3, 4, -5, 7, -6, 8, -4
Dowker-Thistlethwaite code 4 8 14 2 -16 -18 6 -20 -10 -12
Conway Notation [5,21,2-]


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

Length is 10, width is 3,

Braid index is 3

10 125 ML.gif 10 125 AP.gif
[{12, 2}, {1, 10}, {8, 11}, {10, 12}, {9, 3}, {2, 8}, {7, 1}, {6, 9}, {5, 7}, {4, 6}, {3, 5}, {11, 4}]

[edit Notes on presentations of 10 125]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-4][-6]
Hyperbolic Volume 4.61196
A-Polynomial See Data:10 125/A-polynomial

[edit Notes for 10 125's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus 3
Rasmussen s-Invariant 2

[edit Notes for 10 125's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-2 t^2+2 t-1+2 t^{-1} -2 t^{-2} + t^{-3}
Conway polynomial z^6+4 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 11, 2 }
Jones polynomial -q^4+q^3-q^2+2 q-1+2 q^{-1} - q^{-2} + q^{-3} - q^{-4}
HOMFLY-PT polynomial (db, data sources) z^6-a^2 z^4-z^4 a^{-2} +6 z^4-4 a^2 z^2-4 z^2 a^{-2} +11 z^2-3 a^2-3 a^{-2} +7
Kauffman polynomial (db, data sources) a^2 z^8+z^8+a^3 z^7+2 a z^7+z^7 a^{-1} -6 a^2 z^6-6 z^6-6 a^3 z^5-11 a z^5-5 z^5 a^{-1} +11 a^2 z^4+2 z^4 a^{-2} +13 z^4+10 a^3 z^3+17 a z^3+8 z^3 a^{-1} +z^3 a^{-3} -8 a^2 z^2-6 z^2 a^{-2} +z^2 a^{-4} -15 z^2-4 a^3 z-8 a z-6 z a^{-1} -z a^{-3} +z a^{-5} +3 a^2+3 a^{-2} +7
The A2 invariant -q^{12}-q^{10}-q^8+q^4+2 q^2+3+2 q^{-2} + q^{-4} - q^{-8} - q^{-10} - q^{-12}
The G2 invariant q^{60}+q^{56}-q^{54}-q^{48}-2 q^{44}-q^{40}-2 q^{38}-q^{36}-q^{34}-2 q^{32}-2 q^{28}-q^{26}+q^{24}-q^{22}+q^{20}+q^{16}+2 q^{14}+q^{12}+2 q^{10}+2 q^8+2 q^6+2 q^4+3 q^2+1+2 q^{-2} +2 q^{-4} + q^{-6} +2 q^{-8} +2 q^{-10} + q^{-14} +2 q^{-16} - q^{-18} + q^{-20} - q^{-24} + q^{-26} - q^{-28} - q^{-30} - q^{-34} - q^{-36} - q^{-38} -2 q^{-40} - q^{-44} - q^{-46} - q^{-50} - q^{-56} + q^{-72}