10 5

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10 4.gif

10_4

10 6.gif

10_6

Contents

10 5.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X1425 X3,12,4,13 X13,1,14,20 X5,15,6,14 X7,17,8,16 X9,19,10,18 X15,7,16,6 X17,9,18,8 X19,11,20,10 X11,2,12,3
Gauss code -1, 10, -2, 1, -4, 7, -5, 8, -6, 9, -10, 2, -3, 4, -7, 5, -8, 6, -9, 3
Dowker-Thistlethwaite code 4 12 14 16 18 2 20 6 8 10
Conway Notation [6112]


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

Length is 10, width is 3,

Braid index is 3

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

[edit Notes on presentations of 10 5]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 2
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [0][-12]
Hyperbolic Volume 7.37394
A-Polynomial See Data:10 5/A-polynomial

[edit Notes for 10 5's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 5's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-3 t^3+5 t^2-5 t+5-5 t^{-1} +5 t^{-2} -3 t^{-3} + t^{-4}
Conway polynomial z^8+5 z^6+7 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 33, 4 }
Jones polynomial -q^9+2 q^8-3 q^7+4 q^6-5 q^5+5 q^4-4 q^3+4 q^2-2 q+2- q^{-1}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -z^6 a^{-2} +7 z^6 a^{-4} -z^6 a^{-6} -5 z^4 a^{-2} +17 z^4 a^{-4} -5 z^4 a^{-6} -6 z^2 a^{-2} +17 z^2 a^{-4} -7 z^2 a^{-6} - a^{-2} +5 a^{-4} -3 a^{-6}
Kauffman polynomial (db, data sources) z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +4 z^8 a^{-4} +2 z^8 a^{-6} +z^7 a^{-1} -3 z^7 a^{-3} -2 z^7 a^{-5} +2 z^7 a^{-7} -11 z^6 a^{-2} -20 z^6 a^{-4} -7 z^6 a^{-6} +2 z^6 a^{-8} -5 z^5 a^{-1} -2 z^5 a^{-3} -3 z^5 a^{-5} -4 z^5 a^{-7} +2 z^5 a^{-9} +18 z^4 a^{-2} +32 z^4 a^{-4} +10 z^4 a^{-6} -2 z^4 a^{-8} +2 z^4 a^{-10} +6 z^3 a^{-1} +7 z^3 a^{-3} +6 z^3 a^{-5} +3 z^3 a^{-7} -z^3 a^{-9} +z^3 a^{-11} -10 z^2 a^{-2} -22 z^2 a^{-4} -9 z^2 a^{-6} +z^2 a^{-8} -2 z^2 a^{-10} -z a^{-1} -2 z a^{-3} -3 z a^{-5} -z a^{-7} -z a^{-11} + a^{-2} +5 a^{-4} +3 a^{-6}
The A2 invariant -q^2+ q^{-4} +2 q^{-6} + q^{-8} +2 q^{-10} - q^{-12} + q^{-14} - q^{-22} - q^{-26}
The G2 invariant q^{12}-q^{10}+2 q^8-3 q^6+q^4-2 q^2-1+5 q^{-2} -8 q^{-4} +8 q^{-6} -7 q^{-8} +2 q^{-10} +3 q^{-12} -9 q^{-14} +10 q^{-16} -9 q^{-18} +5 q^{-20} +2 q^{-22} -4 q^{-24} +7 q^{-26} -3 q^{-28} +2 q^{-30} +4 q^{-32} -2 q^{-34} +2 q^{-36} +2 q^{-38} -2 q^{-40} +8 q^{-42} -4 q^{-44} +6 q^{-46} -2 q^{-48} - q^{-50} +7 q^{-52} -10 q^{-54} +9 q^{-56} -6 q^{-58} +2 q^{-60} +2 q^{-62} -6 q^{-64} +6 q^{-66} -5 q^{-68} +2 q^{-70} -3 q^{-74} + q^{-76} + q^{-78} -2 q^{-80} + q^{-82} - q^{-86} +2 q^{-88} -2 q^{-90} + q^{-92} + q^{-96} - q^{-98} - q^{-100} - q^{-104} +2 q^{-106} -4 q^{-108} +4 q^{-110} -3 q^{-112} - q^{-114} +2 q^{-116} -5 q^{-118} +5 q^{-120} -5 q^{-122} +3 q^{-124} - q^{-126} -2 q^{-128} +4 q^{-130} -4 q^{-132} +4 q^{-134} -2 q^{-136} + q^{-138} -2 q^{-142} +2 q^{-144} - q^{-146} + q^{-148}