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10 161

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10 160.gif

10_160

10 162.gif

10_162

Contents

10 161.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 161's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_161's page at Knotilus!

Visit 10 161's page at the original Knot Atlas!


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 X1425 X3,12,4,13 X7,14,8,15 X18,9,19,10 X6,19,7,20 X16,5,17,6 X10,17,11,18 X13,8,14,9 X20,15,1,16 X11,2,12,3
Gauss code -1, 10, -2, 1, 6, -5, -3, 8, 4, -7, -10, 2, -8, 3, 9, -6, 7, -4, 5, -9
Dowker-Thistlethwaite code 4 12 -16 14 -18 2 8 -20 -10 -6
Conway Notation [3:-20:-20]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif

Length is 10, width is 3,

Braid index is 3

10 161 ML.gif 10 161 AP.gif
[{11, 3}, {1, 9}, {8, 10}, {9, 11}, {2, 7}, {6, 8}, {7, 4}, {10, 5}, {3, 6}, {4, 1}, {5, 2}]

[edit Notes on presentations of 10 161]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-14][5]
Hyperbolic Volume 5.63877
A-Polynomial See Data:10 161/A-polynomial

[edit Notes for 10 161's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 3
Topological 4 genus 3
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for 10 161's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-2 t+3-2 t^{-1} + t^{-3}
Conway polynomial z^6+6 z^4+7 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 5, -4 }
Jones polynomial  q^{-3} + q^{-6} - q^{-7} + q^{-8} - q^{-9} + q^{-10} - q^{-11}
HOMFLY-PT polynomial (db, data sources) -z^2 a^{10}-a^{10}-z^2 a^8-a^8+z^6 a^6+6 z^4 a^6+9 z^2 a^6+3 a^6
Kauffman polynomial (db, data sources) z^5 a^{13}-4 z^3 a^{13}+3 z a^{13}+z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}+z^5 a^{11}-3 z^3 a^{11}+z a^{11}+z^4 a^{10}-3 z^2 a^{10}+a^{10}-z^4 a^8+3 z^2 a^8-a^8-z^3 a^7+2 z a^7+z^6 a^6-6 z^4 a^6+9 z^2 a^6-3 a^6
The A2 invariant -q^{34}-q^{30}-q^{28}+q^{22}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}
The G2 invariant q^{176}+q^{172}-q^{170}+q^{166}-q^{164}+q^{162}+q^{160}-q^{158}+q^{156}-q^{154}-q^{152}+2 q^{150}-4 q^{148}-2 q^{142}+q^{140}-2 q^{138}-q^{136}-2 q^{132}-q^{130}-q^{126}+q^{124}+q^{118}+q^{116}-q^{112}+2 q^{110}-q^{108}+2 q^{106}-2 q^{102}+2 q^{100}-q^{98}-q^{96}-2 q^{92}+q^{88}-2 q^{86}+2 q^{84}+q^{78}-q^{76}+2 q^{74}+2 q^{72}+q^{70}+q^{68}+q^{66}+q^{64}+2 q^{62}+q^{58}+q^{56}+q^{52}+q^{50}