# 9 40

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 9 40's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 40 at Knotilus!
 In three-fold symmetrical form Symmetrical triangular form (less open) (alternate) Variant Photo of an alsatian chair, musée de l'oeuvre Notre Dame, Strasbourg, 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

Length is 9, width is 4,

Braid index is 4

[{11, 3}, {2, 8}, {9, 4}, {3, 5}, {4, 1}, {7, 2}, {8, 6}, {10, 7}, {5, 9}, {6, 11}, {1, 10}]

### 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

### Four dimensional invariants

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

### 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 $q^{114}-3 q^{112}+6 q^{110}-10 q^{108}+10 q^{106}-8 q^{104}+q^{102}+17 q^{100}-35 q^{98}+57 q^{96}-69 q^{94}+58 q^{92}-26 q^{90}-41 q^{88}+121 q^{86}-182 q^{84}+197 q^{82}-139 q^{80}+14 q^{78}+135 q^{76}-248 q^{74}+274 q^{72}-196 q^{70}+38 q^{68}+122 q^{66}-223 q^{64}+212 q^{62}-79 q^{60}-87 q^{58}+218 q^{56}-237 q^{54}+135 q^{52}+47 q^{50}-232 q^{48}+337 q^{46}-328 q^{44}+209 q^{42}-7 q^{40}-197 q^{38}+334 q^{36}-361 q^{34}+269 q^{32}-104 q^{30}-93 q^{28}+225 q^{26}-263 q^{24}+194 q^{22}-39 q^{20}-123 q^{18}+217 q^{16}-194 q^{14}+58 q^{12}+116 q^{10}-252 q^8+282 q^6-192 q^4+31 q^2+141-245 q^{-2} +261 q^{-4} -179 q^{-6} +57 q^{-8} +53 q^{-10} -121 q^{-12} +126 q^{-14} -87 q^{-16} +44 q^{-18} -2 q^{-20} -19 q^{-22} +24 q^{-24} -20 q^{-26} +10 q^{-28} -4 q^{-30} + q^{-32}$