9 36

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9 35.gif

9_35

9 37.gif

9_37

Contents

9 36.gif
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Knot presentations

Planar diagram presentation X1425 X7,10,8,11 X3948 X9,3,10,2 X11,17,12,16 X5,15,6,14 X15,7,16,6 X13,1,14,18 X17,13,18,12
Gauss code -1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -5, 9, -8, 6, -7, 5, -9, 8
Dowker-Thistlethwaite code 4 8 14 10 2 16 18 6 12
Conway Notation [22,3,2]


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

Length is 9, width is 4,

Braid index is 4

9 36 ML.gif 9 36 AP.gif
[{11, 5}, {6, 4}, {5, 10}, {3, 6}, {8, 11}, {7, 9}, {4, 8}, {2, 7}, {1, 3}, {10, 2}, {9, 1}]

[edit Notes on presentations of 9 36]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index \{4,7\}
Nakanishi index 1
Maximal Thurston-Bennequin number [1][-12]
Hyperbolic Volume 9.88458
A-Polynomial See Data:9 36/A-polynomial

[edit Notes for 9 36'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 36's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+5 t^2-8 t+9-8 t^{-1} +5 t^{-2} - t^{-3}
Conway polynomial -z^6-z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 37, 4 }
Jones polynomial -q^9+2 q^8-4 q^7+6 q^6-6 q^5+6 q^4-5 q^3+4 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} -5 z^2 a^{-4} +6 z^2 a^{-6} -z^2 a^{-8} +2 a^{-2} -3 a^{-4} +4 a^{-6} -2 a^{-8}
Kauffman polynomial (db, data sources) z^8 a^{-4} +z^8 a^{-6} +2 z^7 a^{-3} +5 z^7 a^{-5} +3 z^7 a^{-7} +z^6 a^{-2} +z^6 a^{-4} +4 z^6 a^{-6} +4 z^6 a^{-8} -7 z^5 a^{-3} -14 z^5 a^{-5} -4 z^5 a^{-7} +3 z^5 a^{-9} -4 z^4 a^{-2} -12 z^4 a^{-4} -17 z^4 a^{-6} -7 z^4 a^{-8} +2 z^4 a^{-10} +6 z^3 a^{-3} +9 z^3 a^{-5} -2 z^3 a^{-9} +z^3 a^{-11} +5 z^2 a^{-2} +12 z^2 a^{-4} +15 z^2 a^{-6} +7 z^2 a^{-8} -z^2 a^{-10} -z a^{-3} -2 z a^{-5} +z a^{-7} +z a^{-9} -z a^{-11} -2 a^{-2} -3 a^{-4} -4 a^{-6} -2 a^{-8}
The A2 invariant 1+ q^{-4} + q^{-6} - q^{-8} + q^{-10} -2 q^{-12} + q^{-14} + q^{-16} + q^{-18} +2 q^{-20} - q^{-22} - q^{-26} - q^{-28}
The G2 invariant  q^{-2} - q^{-4} +4 q^{-6} -5 q^{-8} +5 q^{-10} -2 q^{-12} -4 q^{-14} +14 q^{-16} -17 q^{-18} +19 q^{-20} -11 q^{-22} -2 q^{-24} +18 q^{-26} -27 q^{-28} +28 q^{-30} -17 q^{-32} + q^{-34} +13 q^{-36} -23 q^{-38} +20 q^{-40} -9 q^{-42} -5 q^{-44} +15 q^{-46} -18 q^{-48} +9 q^{-50} +3 q^{-52} -18 q^{-54} +24 q^{-56} -24 q^{-58} +16 q^{-60} + q^{-62} -18 q^{-64} +32 q^{-66} -33 q^{-68} +28 q^{-70} -9 q^{-72} -10 q^{-74} +25 q^{-76} -28 q^{-78} +24 q^{-80} -7 q^{-82} -7 q^{-84} +19 q^{-86} -16 q^{-88} +6 q^{-90} +6 q^{-92} -16 q^{-94} +18 q^{-96} -11 q^{-98} -2 q^{-100} +11 q^{-102} -18 q^{-104} +20 q^{-106} -15 q^{-108} +4 q^{-110} +3 q^{-112} -12 q^{-114} +12 q^{-116} -13 q^{-118} +10 q^{-120} -5 q^{-122} + q^{-124} +3 q^{-126} -7 q^{-128} +6 q^{-130} -5 q^{-132} +4 q^{-134} -2 q^{-136} + q^{-140} -2 q^{-142} +2 q^{-144} - q^{-146} + q^{-148}