# 8 4 (KnotPlot image) See the full Rolfsen Knot Table. Visit 8 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 4 at Knotilus!  Somewhat symmetric representation

### Knot presentations

 Planar diagram presentation X6271 X14,10,15,9 X10,3,11,4 X2,13,3,14 X12,5,13,6 X16,8,1,7 X4,11,5,12 X8,16,9,15 Gauss code 1, -4, 3, -7, 5, -1, 6, -8, 2, -3, 7, -5, 4, -2, 8, -6 Dowker-Thistlethwaite code 6 10 12 16 14 4 2 8 Conway Notation 

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index $\{3,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-3] Hyperbolic Volume 5.50049 A-Polynomial See Data:8 4/A-polynomial

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial $-2 t^2+5 t-5+5 t^{-1} -2 t^{-2}$ Conway polynomial $-2 z^4-3 z^2+1$ 2nd Alexander ideal (db, data sources) $\{1\}$ Determinant and Signature { 19, -2 } Jones polynomial $q^3-q^2+2 q-3+3 q^{-1} -3 q^{-2} +3 q^{-3} -2 q^{-4} + q^{-5}$ HOMFLY-PT polynomial (db, data sources) $z^2 a^4+a^4-z^4 a^2-2 z^2 a^2-z^4-3 z^2-2+z^2 a^{-2} +2 a^{-2}$ Kauffman polynomial (db, data sources) $a z^7+z^7 a^{-1} +2 a^2 z^6+z^6 a^{-2} +3 z^6+3 a^3 z^5-a z^5-4 z^5 a^{-1} +3 a^4 z^4-3 a^2 z^4-5 z^4 a^{-2} -11 z^4+2 a^5 z^3-5 a^3 z^3-3 a z^3+4 z^3 a^{-1} +a^6 z^2-3 a^4 z^2-a^2 z^2+7 z^2 a^{-2} +10 z^2+2 a^3 z+a z-z a^{-1} +a^4-2 a^{-2} -2$ The A2 invariant $q^{16}+q^{10}+q^6-q^4-q^2-1- q^{-2} + q^{-4} + q^{-6} + q^{-8} + q^{-10}$ The G2 invariant $q^{86}-q^{84}+q^{82}-q^{80}-q^{74}+3 q^{72}-2 q^{70}+2 q^{68}-2 q^{66}+q^{62}-2 q^{60}+3 q^{58}-2 q^{56}+q^{54}+q^{50}+2 q^{48}-q^{46}+2 q^{44}+2 q^{38}-q^{36}+q^{34}+2 q^{32}-2 q^{30}+3 q^{28}-3 q^{26}+2 q^{22}-6 q^{20}+5 q^{18}-5 q^{16}+2 q^{12}-4 q^{10}+3 q^8-4 q^6+q^4-q^2-2+ q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} - q^{-14} + q^{-16} +3 q^{-18} -4 q^{-20} +4 q^{-22} -2 q^{-24} + q^{-26} +4 q^{-28} -4 q^{-30} +4 q^{-32} - q^{-34} + q^{-36} + q^{-38} -2 q^{-40} +2 q^{-42} + q^{-46}$