10 83

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10 82.gif

10_82

10 84.gif

10_84

Contents

10 83.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

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Warning. There is a mixup in the original (1976) Rolfsen table between the pictures and the invariants of the knots 10_83 and 10_86. That mixup lead to a similar mixup here. In the new (2003) edition of Rolfsen's book the mixup was corrected and on August 17, 2004, it was corrected here (actually in Dror's original Knot Atlas) consistently with Rolfsen's correction. In the years between 1976 and 2003 other authors fixed the problem in different ways and our enumeration here may be different than theirs. Dror would like to thank Z-X. Tao for telling him about the (now corrected) mixup here and A. Stoimenow for telling him about the mixup in Rolfsen's original table.

Knot presentations

Planar diagram presentation X1627 X5,16,6,17 X13,1,14,20 X7,15,8,14 X3948 X9,5,10,4 X19,11,20,10 X11,19,12,18 X17,13,18,12 X15,2,16,3
Gauss code -1, 10, -5, 6, -2, 1, -4, 5, -6, 7, -8, 9, -3, 4, -10, 2, -9, 8, -7, 3
Dowker-Thistlethwaite code 6 8 16 14 4 18 20 2 12 10
Conway Notation [.31.20]


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.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif

Length is 11, width is 4,

Braid index is 4

10 83 ML.gif 10 83 AP.gif
[{10, 4}, {3, 8}, {5, 9}, {4, 7}, {2, 5}, {8, 6}, {7, 11}, {6, 10}, {12, 3}, {11, 13}, {1, 12}, {13, 2}, {9, 1}]

[edit Notes on presentations of 10 83]


Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-2][-10]
Hyperbolic Volume 14.2581
A-Polynomial See Data:10 83/A-polynomial

[edit Notes for 10 83'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 83's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-9 t^2+19 t-23+19 t^{-1} -9 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+3 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 83, 2 }
Jones polynomial -q^8+3 q^7-6 q^6+10 q^5-13 q^4+14 q^3-13 q^2+11 q-7+4 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -z^4+4 z^2 a^{-4} -2 z^2 a^{-6} -z^2- a^{-2} +2 a^{-4} - a^{-6} +1
Kauffman polynomial (db, data sources) 2 z^9 a^{-3} +2 z^9 a^{-5} +5 z^8 a^{-2} +10 z^8 a^{-4} +5 z^8 a^{-6} +6 z^7 a^{-1} +6 z^7 a^{-3} +5 z^7 a^{-5} +5 z^7 a^{-7} -5 z^6 a^{-2} -22 z^6 a^{-4} -10 z^6 a^{-6} +3 z^6 a^{-8} +4 z^6+a z^5-10 z^5 a^{-1} -17 z^5 a^{-3} -18 z^5 a^{-5} -11 z^5 a^{-7} +z^5 a^{-9} -z^4 a^{-2} +22 z^4 a^{-4} +10 z^4 a^{-6} -6 z^4 a^{-8} -7 z^4-a z^3+3 z^3 a^{-1} +13 z^3 a^{-3} +20 z^3 a^{-5} +9 z^3 a^{-7} -2 z^3 a^{-9} -2 z^2 a^{-2} -10 z^2 a^{-4} -4 z^2 a^{-6} +2 z^2 a^{-8} +2 z^2-z a^{-1} -4 z a^{-3} -6 z a^{-5} -3 z a^{-7} + a^{-2} +2 a^{-4} + a^{-6} +1
The A2 invariant -q^6+2 q^4+3 q^{-2} -3 q^{-4} +2 q^{-6} - q^{-8} +2 q^{-12} -2 q^{-14} +3 q^{-16} - q^{-18} - q^{-20} + q^{-22} - q^{-24}
The G2 invariant q^{32}-3 q^{30}+7 q^{28}-13 q^{26}+14 q^{24}-11 q^{22}-2 q^{20}+25 q^{18}-48 q^{16}+71 q^{14}-74 q^{12}+48 q^{10}+3 q^8-73 q^6+144 q^4-179 q^2+164-87 q^{-2} -33 q^{-4} +156 q^{-6} -232 q^{-8} +235 q^{-10} -149 q^{-12} +11 q^{-14} +123 q^{-16} -200 q^{-18} +178 q^{-20} -72 q^{-22} -68 q^{-24} +173 q^{-26} -193 q^{-28} +109 q^{-30} +43 q^{-32} -201 q^{-34} +297 q^{-36} -286 q^{-38} +167 q^{-40} +19 q^{-42} -207 q^{-44} +329 q^{-46} -341 q^{-48} +249 q^{-50} -77 q^{-52} -102 q^{-54} +230 q^{-56} -260 q^{-58} +190 q^{-60} -50 q^{-62} -91 q^{-64} +173 q^{-66} -161 q^{-68} +62 q^{-70} +79 q^{-72} -189 q^{-74} +222 q^{-76} -162 q^{-78} +27 q^{-80} +114 q^{-82} -217 q^{-84} +239 q^{-86} -178 q^{-88} +72 q^{-90} +38 q^{-92} -119 q^{-94} +146 q^{-96} -127 q^{-98} +79 q^{-100} -24 q^{-102} -20 q^{-104} +42 q^{-106} -47 q^{-108} +39 q^{-110} -24 q^{-112} +12 q^{-114} + q^{-116} -7 q^{-118} +7 q^{-120} -7 q^{-122} +4 q^{-124} -2 q^{-126} + q^{-128}