# 8 19 (KnotPlot image) See the full Rolfsen Knot Table. Visit 8 19's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 19 at Knotilus! 8 19 is the first non-obvious torus knot in the table - it is in fact T(4,3). It is also the pretzel knot P(3,3,-2).

8_19 is the first non-homologically thin knot in the Rolfsen table. (That is, it's the first knot whose Khovanov homology has 'off-diagonal' elements.)

### Knot presentations

 Planar diagram presentation X4251 X8493 X9,15,10,14 X5,13,6,12 X13,7,14,6 X11,1,12,16 X15,11,16,10 X2837 Gauss code 1, -8, 2, -1, -4, 5, 8, -2, -3, 7, -6, 4, -5, 3, -7, 6 Dowker-Thistlethwaite code 4 8 -12 2 -14 -16 -6 -10 Conway Notation [3,3,2-]

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 3 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number [-12] Hyperbolic Volume Not hyperbolic A-Polynomial See Data:8 19/A-polynomial

### Four dimensional invariants

 Smooth 4 genus $3$ Topological 4 genus $3$ Concordance genus $3$ Rasmussen s-Invariant -6

### Polynomial invariants

 Alexander polynomial $t^3-t^2+1- t^{-2} + t^{-3}$ Conway polynomial $z^6+5 z^4+5 z^2+1$ 2nd Alexander ideal (db, data sources) $\{1\}$ Determinant and Signature { 3, 6 } Jones polynomial $-q^8+q^5+q^3$ HOMFLY-PT polynomial (db, data sources) $z^6 a^{-6} +6 z^4 a^{-6} -z^4 a^{-8} +10 z^2 a^{-6} -5 z^2 a^{-8} +5 a^{-6} -5 a^{-8} + a^{-10}$ Kauffman polynomial (db, data sources) $z^6 a^{-6} +z^6 a^{-8} +z^5 a^{-7} +z^5 a^{-9} -6 z^4 a^{-6} -6 z^4 a^{-8} -5 z^3 a^{-7} -5 z^3 a^{-9} +10 z^2 a^{-6} +10 z^2 a^{-8} +5 z a^{-7} +5 z a^{-9} -5 a^{-6} -5 a^{-8} - a^{-10}$ The A2 invariant $q^{-10} + q^{-12} +2 q^{-14} +2 q^{-16} +2 q^{-18} - q^{-22} -2 q^{-24} -2 q^{-26} - q^{-28} + q^{-32}$ The G2 invariant $q^{-50} + q^{-52} + q^{-54} + q^{-56} + q^{-58} + q^{-60} +2 q^{-62} +2 q^{-64} + q^{-66} + q^{-68} +2 q^{-70} +2 q^{-72} +2 q^{-74} + q^{-76} + q^{-80} +2 q^{-82} - q^{-94} -2 q^{-96} - q^{-98} - q^{-100} -2 q^{-102} -2 q^{-104} -2 q^{-106} - q^{-108} - q^{-110} -2 q^{-112} -2 q^{-114} - q^{-116} - q^{-122} - q^{-124} + q^{-126} + q^{-128} + q^{-136} + q^{-138} + q^{-144}$