10 163

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10 162.gif

10_162

10 164.gif

10_164

Contents

10 163.gif
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Warning. In 1973 K. Perko noticed that the knots that were later labeled 10161 and 10162 in Rolfsen's tables (which were published in 1976 and were based on earlier tables by Little (1900) and Conway (1970)) are in fact the same. In our table we removed Rolfsen's 10162 and renumbered the subsequent knots, so that our 10 crossings total is 165, one less than Rolfsen's 166. Read more: [1] [2] [3] [4] [5].

Knot presentations

Planar diagram presentation X6271 X8394 X13,19,14,18 X11,1,12,20 X19,13,20,12 X2,16,3,15 X4,17,5,18 X10,6,11,5 X14,7,15,8 X16,10,17,9
Gauss code 1, -6, 2, -7, 8, -1, 9, -2, 10, -8, -4, 5, -3, -9, 6, -10, 7, 3, -5, 4
Dowker-Thistlethwaite code 6 8 10 14 16 -20 -18 2 4 -12
Conway Notation [8*-30]


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

Length is 11, width is 4,

Braid index is 4

10 163 ML.gif 10 163 AP.gif
[{3, 10}, {2, 8}, {1, 4}, {5, 3}, {4, 9}, {7, 2}, {8, 6}, {10, 7}, {9, 5}, {6, 1}]

[edit Notes on presentations of 10 163]


Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-2][-8]
Hyperbolic Volume 13.29
A-Polynomial See Data:10 163/A-polynomial

[edit Notes for 10 163'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 10 163's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-5 t^2+12 t-15+12 t^{-1} -5 t^{-2} + t^{-3}
Conway polynomial z^6+z^4+z^2+1
2nd Alexander ideal (db, data sources) \left\{2,t^2+t+1\right\}
Determinant and Signature { 51, 2 }
Jones polynomial -2 q^6+5 q^5-7 q^4+9 q^3-9 q^2+8 q-6+4 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +3 z^4 a^{-2} -z^4 a^{-4} -z^4+2 z^2 a^{-2} -z^2- a^{-2} +2 a^{-4} - a^{-6} +1
Kauffman polynomial (db, data sources) 2 z^8 a^{-2} +2 z^8 a^{-4} +5 z^7 a^{-1} +8 z^7 a^{-3} +3 z^7 a^{-5} +3 z^6 a^{-2} +z^6 a^{-6} +4 z^6+a z^5-10 z^5 a^{-1} -15 z^5 a^{-3} -4 z^5 a^{-5} -11 z^4 a^{-2} +z^4 a^{-4} +4 z^4 a^{-6} -8 z^4-a z^3+3 z^3 a^{-1} +8 z^3 a^{-3} +7 z^3 a^{-5} +3 z^3 a^{-7} +2 z^2 a^{-2} -4 z^2 a^{-4} -4 z^2 a^{-6} +2 z^2-z a^{-3} -3 z a^{-5} -2 z a^{-7} + a^{-2} +2 a^{-4} + a^{-6} +1
The A2 invariant -q^6+2 q^4+1+2 q^{-2} -3 q^{-4} + q^{-6} -2 q^{-8} +2 q^{-10} +2 q^{-12} +2 q^{-16} -2 q^{-18} - q^{-20}
The G2 invariant q^{32}-3 q^{30}+7 q^{28}-13 q^{26}+13 q^{24}-8 q^{22}-7 q^{20}+31 q^{18}-49 q^{16}+61 q^{14}-48 q^{12}+5 q^{10}+46 q^8-91 q^6+109 q^4-79 q^2+21+50 q^{-2} -92 q^{-4} +96 q^{-6} -53 q^{-8} -13 q^{-10} +68 q^{-12} -90 q^{-14} +59 q^{-16} +6 q^{-18} -74 q^{-20} +115 q^{-22} -108 q^{-24} +61 q^{-26} +6 q^{-28} -81 q^{-30} +123 q^{-32} -134 q^{-34} +101 q^{-36} -34 q^{-38} -40 q^{-40} +99 q^{-42} -113 q^{-44} +90 q^{-46} -30 q^{-48} -35 q^{-50} +79 q^{-52} -79 q^{-54} +39 q^{-56} +32 q^{-58} -85 q^{-60} +106 q^{-62} -70 q^{-64} +3 q^{-66} +62 q^{-68} -104 q^{-70} +100 q^{-72} -65 q^{-74} +13 q^{-76} +32 q^{-78} -59 q^{-80} +55 q^{-82} -36 q^{-84} +14 q^{-86} +5 q^{-88} -14 q^{-90} +9 q^{-92} -8 q^{-94} +5 q^{-96} -2 q^{-98} + q^{-100} + q^{-102}