9 46

From Knot Atlas
Jump to: navigation, search

9 45.gif

9_45

9 47.gif

9_47

Contents

9 46.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 46's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 46 at Knotilus!

9_46 is also known as the pretzel knot P(3,3,-3).


Knot presentations

Planar diagram presentation X4251 X7,12,8,13 X10,3,11,4 X2,11,3,12 X5,14,6,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8
Gauss code 1, -4, 3, -1, -5, 6, -2, 9, -8, -3, 4, 2, -6, 5, -7, 8, -9, 7
Dowker-Thistlethwaite code 4 10 -14 -12 -16 2 -6 -18 -8
Conway Notation [3,3,21-]


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

Length is 9, width is 4,

Braid index is 4

9 46 ML.gif 9 46 AP.gif
[{9, 5}, {3, 8}, {4, 6}, {5, 2}, {1, 4}, {7, 3}, {6, 9}, {2, 7}, {8, 1}]

[edit Notes on presentations of 9 46]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 1
Bridge index 3
Super bridge index 4
Nakanishi index 2
Maximal Thurston-Bennequin number [-7][-1]
Hyperbolic Volume 4.7517
A-Polynomial See Data:9 46/A-polynomial

[edit Notes for 9 46's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 0
Topological 4 genus 0
Concordance genus 0
Rasmussen s-Invariant 0

[edit Notes for 9 46's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t+5-2 t^{-1}
Conway polynomial 1-2 z^2
2nd Alexander ideal (db, data sources) \{3,t+1\}
Determinant and Signature { 9, 0 }
Jones polynomial 2- q^{-1} + q^{-2} -2 q^{-3} + q^{-4} - q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) a^6-z^2 a^4-a^4-z^2 a^2-a^2+2
Kauffman polynomial (db, data sources) a^5 z^7+a^3 z^7+a^6 z^6+2 a^4 z^6+a^2 z^6-5 a^5 z^5-5 a^3 z^5-5 a^6 z^4-9 a^4 z^4-4 a^2 z^4+7 a^5 z^3+8 a^3 z^3+a z^3+6 a^6 z^2+9 a^4 z^2+3 a^2 z^2-4 a^5 z-6 a^3 z-2 a z-a^6-a^4+a^2+2
The A2 invariant q^{20}+q^{18}-q^{12}-q^{10}-q^8-q^6+q^2+2+2 q^{-2}
The G2 invariant q^{94}+q^{90}-q^{88}-q^{82}+2 q^{80}+2 q^{70}+q^{68}-3 q^{66}+q^{62}+2 q^{60}+2 q^{58}-4 q^{56}+3 q^{52}+2 q^{50}-2 q^{48}-5 q^{46}+3 q^{42}+2 q^{40}-4 q^{38}-3 q^{36}+3 q^{32}-5 q^{28}-4 q^{26}+2 q^{24}+2 q^{22}-2 q^{20}-q^{18}-3 q^{16}+3 q^{14}+2 q^{12}-q^{10}+2 q^4+3 q^2+1+ q^{-2} + q^{-4} +3 q^{-8} + q^{-10} - q^{-12} + q^{-14}