# 10 61 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 61's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 61 at Knotilus! 10_61 is also known as the pretzel knot P(4,3,3).

### Knot presentations

 Planar diagram presentation X8291 X10,4,11,3 X2,10,3,9 X18,12,19,11 X14,7,15,8 X16,5,17,6 X6,15,7,16 X4,17,5,18 X20,14,1,13 X12,20,13,19 Gauss code 1, -3, 2, -8, 6, -7, 5, -1, 3, -2, 4, -10, 9, -5, 7, -6, 8, -4, 10, -9 Dowker-Thistlethwaite code 8 10 16 14 2 18 20 6 4 12 Conway Notation [4,3,3]

### Three dimensional invariants

 Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [-1][-11] Hyperbolic Volume 8.45858 A-Polynomial See Data:10 61/A-polynomial

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial $-2 t^3+5 t^2-6 t+7-6 t^{-1} +5 t^{-2} -2 t^{-3}$ Conway polynomial $-2 z^6-7 z^4-4 z^2+1$ 2nd Alexander ideal (db, data sources) $\left\{2,t^2+t+1\right\}$ Determinant and Signature { 33, 4 } Jones polynomial $q^8-2 q^7+3 q^6-4 q^5+5 q^4-5 q^3+4 q^2-4 q+3- q^{-1} + q^{-2}$ HOMFLY-PT polynomial (db, data sources) $-z^6 a^{-2} -z^6 a^{-4} -5 z^4 a^{-2} -4 z^4 a^{-4} +z^4 a^{-6} +z^4-8 z^2 a^{-2} -3 z^2 a^{-4} +3 z^2 a^{-6} +4 z^2-5 a^{-2} + a^{-4} + a^{-6} +4$ Kauffman polynomial (db, data sources) $z^9 a^{-1} +z^9 a^{-3} +4 z^8 a^{-2} +3 z^8 a^{-4} +z^8-5 z^7 a^{-1} +5 z^7 a^{-5} -22 z^6 a^{-2} -10 z^6 a^{-4} +5 z^6 a^{-6} -7 z^6+6 z^5 a^{-1} -16 z^5 a^{-3} -18 z^5 a^{-5} +4 z^5 a^{-7} +38 z^4 a^{-2} +5 z^4 a^{-4} -13 z^4 a^{-6} +3 z^4 a^{-8} +17 z^4+z^3 a^{-1} +26 z^3 a^{-3} +17 z^3 a^{-5} -6 z^3 a^{-7} +2 z^3 a^{-9} -24 z^2 a^{-2} +z^2 a^{-4} +6 z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} -16 z^2-2 z a^{-1} -8 z a^{-3} -6 z a^{-5} +5 a^{-2} + a^{-4} - a^{-6} +4$ The A2 invariant $q^6+q^4+2 q^2+2- q^{-4} -3 q^{-6} -2 q^{-8} +2 q^{-14} + q^{-24}$ The G2 invariant $q^{26}+3 q^{22}-3 q^{20}+3 q^{18}-q^{16}+7 q^{12}-9 q^{10}+10 q^8-3 q^6+q^4+8 q^2-12+11 q^{-2} -2 q^{-4} +5 q^{-8} -9 q^{-10} +4 q^{-12} +5 q^{-14} -4 q^{-16} + q^{-18} -5 q^{-20} - q^{-22} +4 q^{-24} -8 q^{-26} +4 q^{-28} -9 q^{-30} +5 q^{-32} + q^{-34} -7 q^{-36} +6 q^{-38} -12 q^{-40} +8 q^{-42} -5 q^{-44} -3 q^{-46} +7 q^{-48} -9 q^{-50} +5 q^{-52} +3 q^{-54} -3 q^{-56} +4 q^{-58} - q^{-60} -4 q^{-62} +6 q^{-64} -3 q^{-66} +3 q^{-68} + q^{-70} -2 q^{-72} +6 q^{-74} -3 q^{-76} +3 q^{-78} - q^{-80} + q^{-82} - q^{-84} + q^{-86} + q^{-88} -2 q^{-90} +4 q^{-92} -3 q^{-94} +2 q^{-96} - q^{-98} - q^{-100} -3 q^{-104} +3 q^{-106} - q^{-108} + q^{-110} - q^{-114} +2 q^{-116} -2 q^{-118} +2 q^{-120} - q^{-122} - q^{-128} + q^{-130} - q^{-132} + q^{-134}$