10 64

From Knot Atlas
Jump to: navigation, search

10 63.gif

10_63

10 65.gif

10_65

Contents

10 64.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 64's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 64 at Knotilus!


Knot presentations

Planar diagram presentation X8291 X10,4,11,3 X2,10,3,9 X18,12,19,11 X14,5,15,6 X4,17,5,18 X16,7,17,8 X6,15,7,16 X20,14,1,13 X12,20,13,19
Gauss code 1, -3, 2, -6, 5, -8, 7, -1, 3, -2, 4, -10, 9, -5, 8, -7, 6, -4, 10, -9
Dowker-Thistlethwaite code 8 10 14 16 2 18 20 6 4 12
Conway Notation [31,3,3]


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

Length is 10, width is 3,

Braid index is 3

10 64 ML.gif 10 64 AP.gif
[{2, 12}, {1, 11}, {12, 10}, {11, 5}, {9, 4}, {10, 6}, {5, 3}, {4, 2}, {3, 7}, {6, 8}, {7, 9}, {8, 1}]

[edit Notes on presentations of 10 64]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-3][-9]
Hyperbolic Volume 10.8681
A-Polynomial See Data:10 64/A-polynomial

[edit Notes for 10 64's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 64's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+3 t^3-6 t^2+10 t-11+10 t^{-1} -6 t^{-2} +3 t^{-3} - t^{-4}
Conway polynomial -z^8-5 z^6-8 z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 51, 2 }
Jones polynomial q^7-2 q^6+4 q^5-7 q^4+8 q^3-8 q^2+8 q-6+4 q^{-1} -2 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -7 z^6 a^{-2} +z^6 a^{-4} +z^6-18 z^4 a^{-2} +5 z^4 a^{-4} +5 z^4-19 z^2 a^{-2} +8 z^2 a^{-4} +8 z^2-6 a^{-2} +3 a^{-4} +4
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +5 z^8 a^{-2} +3 z^8 a^{-4} +2 z^8+2 a z^7+2 z^7 a^{-3} +4 z^7 a^{-5} +a^2 z^6-18 z^6 a^{-2} -8 z^6 a^{-4} +3 z^6 a^{-6} -6 z^6-7 a z^5-5 z^5 a^{-1} -11 z^5 a^{-3} -11 z^5 a^{-5} +2 z^5 a^{-7} -4 a^2 z^4+30 z^4 a^{-2} +13 z^4 a^{-4} -5 z^4 a^{-6} +z^4 a^{-8} +7 z^4+6 a z^3+4 z^3 a^{-1} +16 z^3 a^{-3} +15 z^3 a^{-5} -3 z^3 a^{-7} +4 a^2 z^2-26 z^2 a^{-2} -8 z^2 a^{-4} +3 z^2 a^{-6} -2 z^2 a^{-8} -9 z^2-a z-3 z a^{-1} -6 z a^{-3} -4 z a^{-5} +6 a^{-2} +3 a^{-4} +4
The A2 invariant q^8+2 q^4+ q^{-2} -2 q^{-4} +2 q^{-6} -2 q^{-8} - q^{-12} - q^{-14} +2 q^{-16} + q^{-20}
The G2 invariant q^{46}-q^{44}+3 q^{42}-5 q^{40}+4 q^{38}-3 q^{36}-2 q^{34}+9 q^{32}-14 q^{30}+20 q^{28}-18 q^{26}+9 q^{24}+5 q^{22}-22 q^{20}+37 q^{18}-41 q^{16}+33 q^{14}-10 q^{12}-16 q^{10}+43 q^8-48 q^6+42 q^4-16 q^2-11+32 q^{-2} -38 q^{-4} +24 q^{-6} +4 q^{-8} -26 q^{-10} +42 q^{-12} -30 q^{-14} +4 q^{-16} +21 q^{-18} -49 q^{-20} +58 q^{-22} -52 q^{-24} +18 q^{-26} +13 q^{-28} -48 q^{-30} +68 q^{-32} -63 q^{-34} +33 q^{-36} -5 q^{-38} -28 q^{-40} +46 q^{-42} -49 q^{-44} +26 q^{-46} +3 q^{-48} -21 q^{-50} +34 q^{-52} -22 q^{-54} -3 q^{-56} +28 q^{-58} -36 q^{-60} +34 q^{-62} -18 q^{-64} -5 q^{-66} +29 q^{-68} -39 q^{-70} +43 q^{-72} -27 q^{-74} +9 q^{-76} +8 q^{-78} -21 q^{-80} +23 q^{-82} -23 q^{-84} +18 q^{-86} -8 q^{-88} - q^{-90} +7 q^{-92} -10 q^{-94} +9 q^{-96} -7 q^{-98} +5 q^{-100} -2 q^{-102} - q^{-104} +2 q^{-106} -3 q^{-108} +2 q^{-110} - q^{-112} + q^{-114}