10 84

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10 83.gif

10_83

10 85.gif

10_85

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Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X8,12,9,11 X20,15,1,16 X16,5,17,6 X12,18,13,17 X14,8,15,7 X18,14,19,13 X6,19,7,20 X2,10,3,9
Gauss code 1, -10, 2, -1, 5, -9, 7, -3, 10, -2, 3, -6, 8, -7, 4, -5, 6, -8, 9, -4
Dowker-Thistlethwaite code 4 10 16 14 2 8 18 20 12 6
Conway Notation [.22.2]


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

Length is 11, width is 4,

Braid index is 4

10 84 ML.gif 10 84 AP.gif
[{3, 11}, {2, 4}, {1, 3}, {7, 2}, {9, 12}, {10, 8}, {6, 9}, {11, 7}, {5, 10}, {4, 6}, {12, 5}, {8, 1}]

[edit Notes on presentations of 10 84]


Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-2][-10]
Hyperbolic Volume 14.7099
A-Polynomial See Data:10 84/A-polynomial

[edit Notes for 10 84's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 84's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-9 t^2+20 t-25+20 t^{-1} -9 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+3 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 87, 2 }
Jones polynomial -q^8+4 q^7-8 q^6+11 q^5-14 q^4+15 q^3-13 q^2+11 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} +2 z^4 a^{-4} -z^4 a^{-6} -z^4+5 z^2 a^{-2} -z^2 a^{-6} -2 z^2+4 a^{-2} -2 a^{-4} -1
Kauffman polynomial (db, data sources) 2 z^9 a^{-3} +2 z^9 a^{-5} +4 z^8 a^{-2} +10 z^8 a^{-4} +6 z^8 a^{-6} +4 z^7 a^{-1} +5 z^7 a^{-3} +8 z^7 a^{-5} +7 z^7 a^{-7} -2 z^6 a^{-2} -17 z^6 a^{-4} -8 z^6 a^{-6} +4 z^6 a^{-8} +3 z^6+a z^5-4 z^5 a^{-1} -11 z^5 a^{-3} -20 z^5 a^{-5} -13 z^5 a^{-7} +z^5 a^{-9} -5 z^4 a^{-2} +9 z^4 a^{-4} +2 z^4 a^{-6} -6 z^4 a^{-8} -6 z^4-2 a z^3-2 z^3 a^{-1} +4 z^3 a^{-3} +11 z^3 a^{-5} +6 z^3 a^{-7} -z^3 a^{-9} +7 z^2 a^{-2} +z^2 a^{-4} -z^2 a^{-6} +z^2 a^{-8} +4 z^2+a z+2 z a^{-1} +2 z a^{-3} -z a^{-7} -4 a^{-2} -2 a^{-4} -1
The A2 invariant -q^6+q^4-q^2-1+4 q^{-2} - q^{-4} +4 q^{-6} + q^{-8} - q^{-10} + q^{-12} -4 q^{-14} +2 q^{-16} - q^{-18} - q^{-20} +2 q^{-22} - q^{-24}
The G2 invariant q^{32}-2 q^{30}+5 q^{28}-8 q^{26}+8 q^{24}-7 q^{22}-2 q^{20}+15 q^{18}-30 q^{16}+43 q^{14}-49 q^{12}+39 q^{10}-14 q^8-31 q^6+88 q^4-135 q^2+155-130 q^{-2} +45 q^{-4} +73 q^{-6} -192 q^{-8} +270 q^{-10} -257 q^{-12} +158 q^{-14} +9 q^{-16} -171 q^{-18} +266 q^{-20} -245 q^{-22} +128 q^{-24} +45 q^{-26} -178 q^{-28} +213 q^{-30} -130 q^{-32} -29 q^{-34} +208 q^{-36} -308 q^{-38} +285 q^{-40} -137 q^{-42} -84 q^{-44} +291 q^{-46} -414 q^{-48} +398 q^{-50} -255 q^{-52} +33 q^{-54} +188 q^{-56} -340 q^{-58} +366 q^{-60} -262 q^{-62} +73 q^{-64} +112 q^{-66} -230 q^{-68} +224 q^{-70} -106 q^{-72} -61 q^{-74} +203 q^{-76} -245 q^{-78} +172 q^{-80} -12 q^{-82} -172 q^{-84} +292 q^{-86} -302 q^{-88} +206 q^{-90} -47 q^{-92} -113 q^{-94} +215 q^{-96} -231 q^{-98} +180 q^{-100} -86 q^{-102} -8 q^{-104} +69 q^{-106} -96 q^{-108} +82 q^{-110} -51 q^{-112} +23 q^{-114} +2 q^{-116} -13 q^{-118} +15 q^{-120} -13 q^{-122} +7 q^{-124} -3 q^{-126} + q^{-128}