# 9 31 (KnotPlot image) See the full Rolfsen Knot Table. Visit 9 31's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 31 at Knotilus!

### Knot presentations

 Planar diagram presentation X1425 X3,10,4,11 X11,1,12,18 X5,13,6,12 X17,7,18,6 X7,14,8,15 X13,16,14,17 X15,8,16,9 X9,2,10,3 Gauss code -1, 9, -2, 1, -4, 5, -6, 8, -9, 2, -3, 4, -7, 6, -8, 7, -5, 3 Dowker-Thistlethwaite code 4 10 12 14 2 18 16 8 6 Conway Notation 

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-9][-2] Hyperbolic Volume 11.6863 A-Polynomial See Data:9 31/A-polynomial

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial $t^3-5 t^2+13 t-17+13 t^{-1} -5 t^{-2} + t^{-3}$ Conway polynomial $z^6+z^4+2 z^2+1$ 2nd Alexander ideal (db, data sources) $\{1\}$ Determinant and Signature { 55, -2 } Jones polynomial $-q^2+3 q-5+8 q^{-1} -9 q^{-2} +10 q^{-3} -8 q^{-4} +6 q^{-5} -4 q^{-6} + q^{-7}$ HOMFLY-PT polynomial (db, data sources) $z^2 a^6-2 z^4 a^4-4 z^2 a^4-2 a^4+z^6 a^2+4 z^4 a^2+7 z^2 a^2+4 a^2-z^4-2 z^2-1$ Kauffman polynomial (db, data sources) $z^4 a^8+4 z^5 a^7-4 z^3 a^7+6 z^6 a^6-8 z^4 a^6+3 z^2 a^6+4 z^7 a^5+z^5 a^5-8 z^3 a^5+3 z a^5+z^8 a^4+11 z^6 a^4-23 z^4 a^4+13 z^2 a^4-2 a^4+7 z^7 a^3-7 z^5 a^3-5 z^3 a^3+5 z a^3+z^8 a^2+8 z^6 a^2-21 z^4 a^2+15 z^2 a^2-4 a^2+3 z^7 a-3 z^5 a-3 z^3 a+3 z a+3 z^6-7 z^4+5 z^2-1+z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1}$ The A2 invariant $q^{22}-q^{20}-2 q^{18}+q^{16}-2 q^{14}+q^{12}+q^{10}+3 q^6-q^4+3 q^2- q^{-2} + q^{-4} - q^{-6}$ The G2 invariant $q^{114}-3 q^{112}+6 q^{110}-10 q^{108}+8 q^{106}-4 q^{104}-5 q^{102}+22 q^{100}-33 q^{98}+45 q^{96}-41 q^{94}+16 q^{92}+17 q^{90}-54 q^{88}+80 q^{86}-86 q^{84}+65 q^{82}-20 q^{80}-33 q^{78}+75 q^{76}-90 q^{74}+70 q^{72}-28 q^{70}-23 q^{68}+48 q^{66}-52 q^{64}+24 q^{62}+26 q^{60}-67 q^{58}+82 q^{56}-60 q^{54}+2 q^{52}+65 q^{50}-121 q^{48}+136 q^{46}-108 q^{44}+48 q^{42}+32 q^{40}-96 q^{38}+129 q^{36}-115 q^{34}+68 q^{32}-4 q^{30}-51 q^{28}+73 q^{26}-55 q^{24}+22 q^{22}+29 q^{20}-57 q^{18}+59 q^{16}-26 q^{14}-24 q^{12}+72 q^{10}-94 q^8+83 q^6-43 q^4-9 q^2+55-80 q^{-2} +81 q^{-4} -55 q^{-6} +18 q^{-8} +13 q^{-10} -35 q^{-12} +38 q^{-14} -31 q^{-16} +19 q^{-18} -5 q^{-20} -5 q^{-22} +7 q^{-24} -8 q^{-26} +5 q^{-28} -2 q^{-30} + q^{-32}$