10 51

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10 50.gif

10_50

10 52.gif

10_52

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Knot presentations

Planar diagram presentation X1425 X3849 X9,17,10,16 X5,15,6,14 X15,7,16,6 X13,1,14,20 X19,11,20,10 X11,19,12,18 X17,13,18,12 X7283
Gauss code -1, 10, -2, 1, -4, 5, -10, 2, -3, 7, -8, 9, -6, 4, -5, 3, -9, 8, -7, 6
Dowker-Thistlethwaite code 4 8 14 2 16 18 20 6 12 10
Conway Notation [32,21,2]


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.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif

Length is 11, width is 4,

Braid index is 4

10 51 ML.gif 10 51 AP.gif
[{9, 4}, {3, 7}, {6, 8}, {7, 9}, {8, 11}, {5, 10}, {4, 6}, {2, 5}, {12, 3}, {11, 13}, {1, 12}, {13, 2}, {10, 1}]

[edit Notes on presentations of 10 51]


Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-2][-10]
Hyperbolic Volume 12.6314
A-Polynomial See Data:10 51/A-polynomial

[edit Notes for 10 51's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 51's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-7 t^2+15 t-19+15 t^{-1} -7 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+5 z^4+5 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 67, 2 }
Jones polynomial -q^8+2 q^7-5 q^6+8 q^5-10 q^4+12 q^3-10 q^2+9 q-6+3 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6 a^{-4} +3 z^4 a^{-2} +4 z^4 a^{-4} -z^4 a^{-6} -z^4+3 z^2 a^{-2} +7 z^2 a^{-4} -3 z^2 a^{-6} -2 z^2+ a^{-2} +4 a^{-4} -3 a^{-6} -1
Kauffman polynomial (db, data sources) z^9 a^{-3} +z^9 a^{-5} +3 z^8 a^{-2} +6 z^8 a^{-4} +3 z^8 a^{-6} +4 z^7 a^{-1} +6 z^7 a^{-3} +5 z^7 a^{-5} +3 z^7 a^{-7} -z^6 a^{-2} -12 z^6 a^{-4} -6 z^6 a^{-6} +2 z^6 a^{-8} +3 z^6+a z^5-6 z^5 a^{-1} -16 z^5 a^{-3} -16 z^5 a^{-5} -6 z^5 a^{-7} +z^5 a^{-9} -6 z^4 a^{-2} +13 z^4 a^{-4} +9 z^4 a^{-6} -4 z^4 a^{-8} -6 z^4-2 a z^3+15 z^3 a^{-3} +21 z^3 a^{-5} +5 z^3 a^{-7} -3 z^3 a^{-9} +4 z^2 a^{-2} -8 z^2 a^{-4} -8 z^2 a^{-6} +z^2 a^{-8} +3 z^2+a z-5 z a^{-3} -9 z a^{-5} -3 z a^{-7} +2 z a^{-9} - a^{-2} +4 a^{-4} +3 a^{-6} -1
The A2 invariant -q^6+q^4-q^2-1+2 q^{-2} -2 q^{-4} +3 q^{-6} + q^{-8} +2 q^{-10} +3 q^{-12} - q^{-14} +2 q^{-16} -2 q^{-18} -2 q^{-20} - q^{-24}
The G2 invariant q^{32}-2 q^{30}+5 q^{28}-8 q^{26}+8 q^{24}-6 q^{22}-3 q^{20}+16 q^{18}-31 q^{16}+42 q^{14}-44 q^{12}+24 q^{10}+7 q^8-48 q^6+87 q^4-101 q^2+88-42 q^{-2} -28 q^{-4} +91 q^{-6} -127 q^{-8} +120 q^{-10} -70 q^{-12} -2 q^{-14} +67 q^{-16} -98 q^{-18} +82 q^{-20} -25 q^{-22} -44 q^{-24} +92 q^{-26} -95 q^{-28} +44 q^{-30} +39 q^{-32} -116 q^{-34} +163 q^{-36} -147 q^{-38} +84 q^{-40} +20 q^{-42} -116 q^{-44} +180 q^{-46} -180 q^{-48} +130 q^{-50} -33 q^{-52} -57 q^{-54} +120 q^{-56} -126 q^{-58} +89 q^{-60} -17 q^{-62} -50 q^{-64} +83 q^{-66} -73 q^{-68} +18 q^{-70} +51 q^{-72} -103 q^{-74} +113 q^{-76} -78 q^{-78} +6 q^{-80} +62 q^{-82} -114 q^{-84} +123 q^{-86} -94 q^{-88} +37 q^{-90} +16 q^{-92} -60 q^{-94} +74 q^{-96} -66 q^{-98} +43 q^{-100} -15 q^{-102} -7 q^{-104} +19 q^{-106} -24 q^{-108} +20 q^{-110} -14 q^{-112} +8 q^{-114} - q^{-116} -3 q^{-118} +4 q^{-120} -4 q^{-122} +3 q^{-124} - q^{-126} + q^{-128}