# 10 81 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 81's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 81 at Knotilus!

### Knot presentations

 Planar diagram presentation X4251 X8493 X12,6,13,5 X16,9,17,10 X20,17,1,18 X18,13,19,14 X14,19,15,20 X10,15,11,16 X6,12,7,11 X2837 Gauss code 1, -10, 2, -1, 3, -9, 10, -2, 4, -8, 9, -3, 6, -7, 8, -4, 5, -6, 7, -5 Dowker-Thistlethwaite code 4 8 12 2 16 6 18 10 20 14 Conway Notation [(21,2)(21,2)]

### Three dimensional invariants

 Symmetry type Negative amphicheiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 14.4927 A-Polynomial See Data:10 81/A-polynomial

### Four dimensional invariants

 Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial $-t^3+8 t^2-20 t+27-20 t^{-1} +8 t^{-2} - t^{-3}$ Conway polynomial $-z^6+2 z^4+3 z^2+1$ 2nd Alexander ideal (db, data sources) $\{1\}$ Determinant and Signature { 85, 0 } Jones polynomial $-q^5+3 q^4-7 q^3+11 q^2-13 q+15-13 q^{-1} +11 q^{-2} -7 q^{-3} +3 q^{-4} - q^{-5}$ HOMFLY-PT polynomial (db, data sources) $-z^6+2 a^2 z^4+2 z^4 a^{-2} -2 z^4-a^4 z^2+3 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} -z^2-a^4+a^2+ a^{-2} - a^{-4} +1$ Kauffman polynomial (db, data sources) $a z^9+z^9 a^{-1} +4 a^2 z^8+4 z^8 a^{-2} +8 z^8+5 a^3 z^7+13 a z^7+13 z^7 a^{-1} +5 z^7 a^{-3} +3 a^4 z^6+3 z^6 a^{-4} -6 z^6+a^5 z^5-8 a^3 z^5-31 a z^5-31 z^5 a^{-1} -8 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4-9 a^2 z^4-9 z^4 a^{-2} -5 z^4 a^{-4} -8 z^4-2 a^5 z^3+5 a^3 z^3+25 a z^3+25 z^3 a^{-1} +5 z^3 a^{-3} -2 z^3 a^{-5} +3 a^4 z^2+6 a^2 z^2+6 z^2 a^{-2} +3 z^2 a^{-4} +6 z^2+a^5 z-2 a^3 z-8 a z-8 z a^{-1} -2 z a^{-3} +z a^{-5} -a^4-a^2- a^{-2} - a^{-4} +1$ The A2 invariant $-q^{16}+q^{12}-3 q^{10}+2 q^8-q^4+4 q^2-1+4 q^{-2} - q^{-4} +2 q^{-8} -3 q^{-10} + q^{-12} - q^{-16}$ The G2 invariant $q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+9 q^{72}-9 q^{70}+q^{68}+14 q^{66}-35 q^{64}+56 q^{62}-69 q^{60}+55 q^{58}-16 q^{56}-50 q^{54}+129 q^{52}-183 q^{50}+191 q^{48}-130 q^{46}+4 q^{44}+139 q^{42}-255 q^{40}+293 q^{38}-227 q^{36}+79 q^{34}+91 q^{32}-219 q^{30}+247 q^{28}-166 q^{26}+14 q^{24}+137 q^{22}-214 q^{20}+173 q^{18}-34 q^{16}-147 q^{14}+296 q^{12}-335 q^{10}+249 q^8-55 q^6-172 q^4+360 q^2-427+360 q^{-2} -172 q^{-4} -55 q^{-6} +249 q^{-8} -335 q^{-10} +296 q^{-12} -147 q^{-14} -34 q^{-16} +173 q^{-18} -214 q^{-20} +137 q^{-22} +14 q^{-24} -166 q^{-26} +247 q^{-28} -219 q^{-30} +91 q^{-32} +79 q^{-34} -227 q^{-36} +293 q^{-38} -255 q^{-40} +139 q^{-42} +4 q^{-44} -130 q^{-46} +191 q^{-48} -183 q^{-50} +129 q^{-52} -50 q^{-54} -16 q^{-56} +55 q^{-58} -69 q^{-60} +56 q^{-62} -35 q^{-64} +14 q^{-66} + q^{-68} -9 q^{-70} +9 q^{-72} -8 q^{-74} +5 q^{-76} -2 q^{-78} + q^{-80}$