9 40

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9 39.gif

9_39

9 41.gif

9_41

Contents

9 40.gif
(KnotPlot image)

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In three-fold symmetrical form
Symmetrical triangular form
(less open)
(alternate)
Variant
Photo of an alsatian chair, France.
Cylindrical depiction.

Knot presentations

Planar diagram presentation X1627 X7,12,8,13 X5,15,6,14 X11,3,12,2 X15,10,16,11 X3,16,4,17 X9,4,10,5 X17,9,18,8 X13,18,14,1
Gauss code -1, 4, -6, 7, -3, 1, -2, 8, -7, 5, -4, 2, -9, 3, -5, 6, -8, 9
Dowker-Thistlethwaite code 6 16 14 12 4 2 18 10 8
Conway Notation [9*]


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

Length is 9, width is 4,

Braid index is 4

9 40 ML.gif 9 40 AP.gif
[{11, 3}, {2, 8}, {9, 4}, {3, 5}, {4, 1}, {7, 2}, {8, 6}, {10, 7}, {5, 9}, {6, 11}, {1, 10}]

[edit Notes on presentations of 9 40]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index 4
Nakanishi index 2
Maximal Thurston-Bennequin number [-9][-2]
Hyperbolic Volume 15.0183
A-Polynomial See Data:9 40/A-polynomial

[edit Notes for 9 40's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus [1,3]
Rasmussen s-Invariant -2

[edit Notes for 9 40's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-7 t^2+18 t-23+18 t^{-1} -7 t^{-2} + t^{-3}
Conway polynomial z^6-z^4-z^2+1
2nd Alexander ideal (db, data sources) \left\{t^2-3 t+1\right\}
Determinant and Signature { 75, -2 }
Jones polynomial -q^2+5 q-8+11 q^{-1} -13 q^{-2} +13 q^{-3} -11 q^{-4} +8 q^{-5} -4 q^{-6} + q^{-7}
HOMFLY-PT polynomial (db, data sources) z^2 a^6-2 z^4 a^4-2 z^2 a^4+a^4+z^6 a^2+2 z^4 a^2-2 a^2-z^4+2
Kauffman polynomial (db, data sources) z^4 a^8+4 z^5 a^7-2 z^3 a^7+8 z^6 a^6-9 z^4 a^6+4 z^2 a^6+9 z^7 a^5-12 z^5 a^5+6 z^3 a^5-z a^5+4 z^8 a^4+7 z^6 a^4-20 z^4 a^4+7 z^2 a^4+a^4+17 z^7 a^3-32 z^5 a^3+14 z^3 a^3-z a^3+4 z^8 a^2+4 z^6 a^2-17 z^4 a^2+3 z^2 a^2+2 a^2+8 z^7 a-15 z^5 a+6 z^3 a+5 z^6-7 z^4+2+z^5 a^{-1}
The A2 invariant q^{22}-q^{20}-2 q^{18}+3 q^{16}-q^{14}+2 q^{12}+q^{10}-3 q^8+q^6-4 q^4+3 q^2+1+3 q^{-4} - q^{-6}
The G2 invariant q^{114}-3 q^{112}+6 q^{110}-10 q^{108}+10 q^{106}-8 q^{104}+q^{102}+17 q^{100}-35 q^{98}+57 q^{96}-69 q^{94}+58 q^{92}-26 q^{90}-41 q^{88}+121 q^{86}-182 q^{84}+197 q^{82}-139 q^{80}+14 q^{78}+135 q^{76}-248 q^{74}+274 q^{72}-196 q^{70}+38 q^{68}+122 q^{66}-223 q^{64}+212 q^{62}-79 q^{60}-87 q^{58}+218 q^{56}-237 q^{54}+135 q^{52}+47 q^{50}-232 q^{48}+337 q^{46}-328 q^{44}+209 q^{42}-7 q^{40}-197 q^{38}+334 q^{36}-361 q^{34}+269 q^{32}-104 q^{30}-93 q^{28}+225 q^{26}-263 q^{24}+194 q^{22}-39 q^{20}-123 q^{18}+217 q^{16}-194 q^{14}+58 q^{12}+116 q^{10}-252 q^8+282 q^6-192 q^4+31 q^2+141-245 q^{-2} +261 q^{-4} -179 q^{-6} +57 q^{-8} +53 q^{-10} -121 q^{-12} +126 q^{-14} -87 q^{-16} +44 q^{-18} -2 q^{-20} -19 q^{-22} +24 q^{-24} -20 q^{-26} +10 q^{-28} -4 q^{-30} + q^{-32}