10 50

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10 49.gif

10_49

10 51.gif

10_51

Contents

10 50.gif
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Knot presentations

Planar diagram presentation X6271 X8493 X2837 X16,10,17,9 X14,5,15,6 X4,15,5,16 X20,14,1,13 X10,20,11,19 X18,12,19,11 X12,18,13,17
Gauss code 1, -3, 2, -6, 5, -1, 3, -2, 4, -8, 9, -10, 7, -5, 6, -4, 10, -9, 8, -7
Dowker-Thistlethwaite code 6 8 14 2 16 18 20 4 12 10
Conway Notation [32,3,2]


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

Length is 11, width is 4,

Braid index is 4

10 50 ML.gif 10 50 AP.gif
[{2, 12}, {1, 11}, {12, 10}, {11, 7}, {9, 4}, {10, 8}, {5, 3}, {4, 6}, {7, 5}, {6, 2}, {3, 9}, {8, 1}]

[edit Notes on presentations of 10 50]


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 [1][-13]
Hyperbolic Volume 11.1989
A-Polynomial See Data:10 50/A-polynomial

[edit Notes for 10 50's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 50's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+7 t^2-11 t+13-11 t^{-1} +7 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-5 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 53, 4 }
Jones polynomial q^{10}-2 q^9+4 q^8-7 q^7+8 q^6-9 q^5+8 q^4-6 q^3+5 q^2-2 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -3 z^4 a^{-4} -4 z^4 a^{-6} +z^4 a^{-8} +3 z^2 a^{-2} -z^2 a^{-4} -6 z^2 a^{-6} +3 z^2 a^{-8} +2 a^{-2} + a^{-4} -4 a^{-6} +2 a^{-8}
Kauffman polynomial (db, data sources) z^9 a^{-5} +z^9 a^{-7} +2 z^8 a^{-4} +5 z^8 a^{-6} +3 z^8 a^{-8} +2 z^7 a^{-3} +z^7 a^{-5} +3 z^7 a^{-7} +4 z^7 a^{-9} +z^6 a^{-2} -4 z^6 a^{-4} -15 z^6 a^{-6} -7 z^6 a^{-8} +3 z^6 a^{-10} -6 z^5 a^{-3} -8 z^5 a^{-5} -15 z^5 a^{-7} -11 z^5 a^{-9} +2 z^5 a^{-11} -4 z^4 a^{-2} -z^4 a^{-4} +18 z^4 a^{-6} +9 z^4 a^{-8} -5 z^4 a^{-10} +z^4 a^{-12} +3 z^3 a^{-3} +6 z^3 a^{-5} +22 z^3 a^{-7} +16 z^3 a^{-9} -3 z^3 a^{-11} +5 z^2 a^{-2} -13 z^2 a^{-6} -3 z^2 a^{-8} +3 z^2 a^{-10} -2 z^2 a^{-12} +z a^{-3} -3 z a^{-5} -10 z a^{-7} -6 z a^{-9} -2 a^{-2} + a^{-4} +4 a^{-6} +2 a^{-8}
The A2 invariant 1+ q^{-4} +2 q^{-6} +3 q^{-10} - q^{-12} - q^{-16} -3 q^{-18} -2 q^{-22} + q^{-24} + q^{-26} + q^{-30}
The G2 invariant  q^{-2} - q^{-4} +4 q^{-6} -5 q^{-8} +6 q^{-10} -3 q^{-12} -2 q^{-14} +12 q^{-16} -18 q^{-18} +25 q^{-20} -23 q^{-22} +11 q^{-24} +9 q^{-26} -30 q^{-28} +49 q^{-30} -48 q^{-32} +36 q^{-34} -7 q^{-36} -25 q^{-38} +51 q^{-40} -55 q^{-42} +41 q^{-44} -8 q^{-46} -22 q^{-48} +41 q^{-50} -37 q^{-52} +15 q^{-54} +17 q^{-56} -40 q^{-58} +46 q^{-60} -32 q^{-62} - q^{-64} +35 q^{-66} -64 q^{-68} +68 q^{-70} -53 q^{-72} +15 q^{-74} +23 q^{-76} -62 q^{-78} +71 q^{-80} -64 q^{-82} +32 q^{-84} +4 q^{-86} -40 q^{-88} +51 q^{-90} -43 q^{-92} +16 q^{-94} +15 q^{-96} -34 q^{-98} +34 q^{-100} -14 q^{-102} -12 q^{-104} +38 q^{-106} -43 q^{-108} +38 q^{-110} -13 q^{-112} -13 q^{-114} +35 q^{-116} -42 q^{-118} +41 q^{-120} -24 q^{-122} +6 q^{-124} +10 q^{-126} -21 q^{-128} +23 q^{-130} -22 q^{-132} +16 q^{-134} -7 q^{-136} - q^{-138} +6 q^{-140} -10 q^{-142} +9 q^{-144} -7 q^{-146} +5 q^{-148} -2 q^{-150} - q^{-152} +2 q^{-154} -3 q^{-156} +2 q^{-158} - q^{-160} + q^{-162}