10 85

From Knot Atlas
Jump to: navigation, search

10 84.gif

10_84

10 86.gif

10_86

Contents

10 85.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 85's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 85 at Knotilus!


Knot presentations

Planar diagram presentation X6271 X16,6,17,5 X18,11,19,12 X14,7,15,8 X8394 X4,9,5,10 X20,13,1,14 X10,17,11,18 X12,19,13,20 X2,16,3,15
Gauss code 1, -10, 5, -6, 2, -1, 4, -5, 6, -8, 3, -9, 7, -4, 10, -2, 8, -3, 9, -7
Dowker-Thistlethwaite code 6 8 16 14 4 18 20 2 10 12
Conway Notation [.4.20]


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

Length is 10, width is 3,

Braid index is 3

10 85 ML.gif 10 85 AP.gif
[{3, 12}, {2, 6}, {8, 13}, {9, 7}, {10, 8}, {11, 9}, {4, 10}, {6, 11}, {5, 3}, {12, 4}, {1, 5}, {13, 2}, {7, 1}]

[edit Notes on presentations of 10 85]


Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-12][0]
Hyperbolic Volume 11.7978
A-Polynomial See Data:10 85/A-polynomial

[edit Notes for 10 85's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 85's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-4 t^3+8 t^2-10 t+11-10 t^{-1} +8 t^{-2} -4 t^{-3} + t^{-4}
Conway polynomial z^8+4 z^6+4 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 57, -4 }
Jones polynomial -q+3-4 q^{-1} +7 q^{-2} -8 q^{-3} +9 q^{-4} -9 q^{-5} +7 q^{-6} -5 q^{-7} +3 q^{-8} - q^{-9}
HOMFLY-PT polynomial (db, data sources) a^4 z^8-a^6 z^6+6 a^4 z^6-a^2 z^6-4 a^6 z^4+12 a^4 z^4-4 a^2 z^4-4 a^6 z^2+9 a^4 z^2-3 a^2 z^2-a^6+a^4+a^2
Kauffman polynomial (db, data sources) z^3 a^{11}+3 z^4 a^{10}-z^2 a^{10}+5 z^5 a^9-4 z^3 a^9+z a^9+6 z^6 a^8-7 z^4 a^8+z^2 a^8+6 z^7 a^7-10 z^5 a^7+2 z^3 a^7+5 z^8 a^6-12 z^6 a^6+8 z^4 a^6-5 z^2 a^6+a^6+2 z^9 a^5-15 z^5 a^5+14 z^3 a^5-2 z a^5+8 z^8 a^4-32 z^6 a^4+37 z^4 a^4-14 z^2 a^4+a^4+2 z^9 a^3-5 z^7 a^3-4 z^5 a^3+11 z^3 a^3-2 z a^3+3 z^8 a^2-14 z^6 a^2+19 z^4 a^2-7 z^2 a^2-a^2+z^7 a-4 z^5 a+4 z^3 a-z a
The A2 invariant -q^{26}+q^{24}-q^{22}+q^{20}-q^{16}+q^{14}-3 q^{12}+2 q^{10}+2 q^6+2 q^4+1- q^{-2}
The G2 invariant q^{148}-2 q^{146}+3 q^{144}-4 q^{142}+2 q^{140}-q^{138}-2 q^{136}+8 q^{134}-11 q^{132}+14 q^{130}-12 q^{128}+6 q^{126}+2 q^{124}-11 q^{122}+20 q^{120}-24 q^{118}+21 q^{116}-14 q^{114}+q^{112}+9 q^{110}-16 q^{108}+22 q^{106}-24 q^{104}+21 q^{102}-17 q^{100}+3 q^{98}+12 q^{96}-26 q^{94}+33 q^{92}-29 q^{90}+14 q^{88}+9 q^{86}-28 q^{84}+33 q^{82}-18 q^{80}-7 q^{78}+34 q^{76}-45 q^{74}+30 q^{72}+5 q^{70}-40 q^{68}+64 q^{66}-65 q^{64}+40 q^{62}-2 q^{60}-38 q^{58}+62 q^{56}-69 q^{54}+51 q^{52}-22 q^{50}-13 q^{48}+37 q^{46}-47 q^{44}+43 q^{42}-20 q^{40}-10 q^{38}+33 q^{36}-39 q^{34}+27 q^{32}+7 q^{30}-38 q^{28}+60 q^{26}-49 q^{24}+21 q^{22}+22 q^{20}-54 q^{18}+68 q^{16}-53 q^{14}+23 q^{12}+9 q^{10}-34 q^8+42 q^6-33 q^4+18 q^2-2-8 q^{-2} +9 q^{-4} -9 q^{-6} +5 q^{-8} -2 q^{-10} + q^{-12}