9 11

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9 10.gif

9_10

9 12.gif

9_12

Contents

9 11.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X13,1,14,18 X5,15,6,14 X7,17,8,16 X15,7,16,6 X17,9,18,8
Gauss code -1, 4, -3, 1, -6, 8, -7, 9, -2, 3, -4, 2, -5, 6, -8, 7, -9, 5
Dowker-Thistlethwaite code 4 10 14 16 12 2 18 6 8
Conway Notation [4122]


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

Length is 9, width is 4,

Braid index is 4

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

[edit Notes on presentations of 9 11]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index \{4,6\}
Nakanishi index 1
Maximal Thurston-Bennequin number [1][-12]
Hyperbolic Volume 8.28859
A-Polynomial See Data:9 11/A-polynomial

[edit Notes for 9 11'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 9 11's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+5 t^2-7 t+7-7 t^{-1} +5 t^{-2} - t^{-3}
Conway polynomial -z^6-z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 33, 4 }
Jones polynomial -q^9+2 q^8-4 q^7+5 q^6-5 q^5+6 q^4-4 q^3+3 q^2-2 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} +z^4 a^{-2} -4 z^4 a^{-4} +2 z^4 a^{-6} +3 z^2 a^{-2} -4 z^2 a^{-4} +6 z^2 a^{-6} -z^2 a^{-8} + a^{-2} - a^{-4} +3 a^{-6} -2 a^{-8}
Kauffman polynomial (db, data sources) z^8 a^{-4} +z^8 a^{-6} +2 z^7 a^{-3} +4 z^7 a^{-5} +2 z^7 a^{-7} +z^6 a^{-2} -z^6 a^{-4} +z^6 a^{-6} +3 z^6 a^{-8} -8 z^5 a^{-3} -12 z^5 a^{-5} -z^5 a^{-7} +3 z^5 a^{-9} -4 z^4 a^{-2} -5 z^4 a^{-4} -7 z^4 a^{-6} -4 z^4 a^{-8} +2 z^4 a^{-10} +8 z^3 a^{-3} +9 z^3 a^{-5} -3 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} +4 z^2 a^{-2} +5 z^2 a^{-4} +6 z^2 a^{-6} +4 z^2 a^{-8} -z^2 a^{-10} -z a^{-3} -2 z a^{-5} +2 z a^{-7} +2 z a^{-9} -z a^{-11} - a^{-2} - a^{-4} -3 a^{-6} -2 a^{-8}
The A2 invariant 1- q^{-8} +2 q^{-10} +2 q^{-14} + q^{-16} + q^{-20} - q^{-22} - q^{-26} - q^{-28}
The G2 invariant  q^{-2} - q^{-4} +3 q^{-6} -4 q^{-8} +3 q^{-10} - q^{-12} -2 q^{-14} +10 q^{-16} -12 q^{-18} +13 q^{-20} -7 q^{-22} -2 q^{-24} +10 q^{-26} -17 q^{-28} +17 q^{-30} -12 q^{-32} +2 q^{-34} +8 q^{-36} -13 q^{-38} +12 q^{-40} -7 q^{-42} -2 q^{-44} +7 q^{-46} -8 q^{-48} +4 q^{-50} - q^{-52} -5 q^{-54} +16 q^{-56} -12 q^{-58} +10 q^{-60} - q^{-62} -8 q^{-64} +19 q^{-66} -20 q^{-68} +18 q^{-70} -9 q^{-72} +16 q^{-76} -19 q^{-78} +17 q^{-80} -8 q^{-82} -2 q^{-84} +8 q^{-86} -10 q^{-88} +4 q^{-90} -4 q^{-94} +7 q^{-96} -6 q^{-98} +3 q^{-102} -10 q^{-104} +10 q^{-106} -9 q^{-108} +4 q^{-110} - q^{-112} -4 q^{-114} +8 q^{-116} -11 q^{-118} +11 q^{-120} -7 q^{-122} +2 q^{-124} + q^{-126} -6 q^{-128} +6 q^{-130} -6 q^{-132} +5 q^{-134} -2 q^{-136} + q^{-140} -2 q^{-142} +2 q^{-144} - q^{-146} + q^{-148}