Symplectic quandle method and SL(2,ℂ)-representations of 2-bridge Knots

In this paper, we extend the symplectic quandle method, previously employed in our study of parabolic representations of knot groups, to investigate the general [Formula: see text]-representations of 2-bridge “kmot” groups. We introduce a ‘generalized symplectic quandle structure’ corresponding to ([Formula: see text], conjugation) for each [Formula: see text], where [Formula: see text]. By converting the system of conjugation quandle equations to that of generalized symplectic quandle equations, we obtain a simpler expression for the 2-variable Riley polynomial and derive some recursive formulas for Riley polynomials and Alexander polynomials. This approach enables us to effectively compute the A-polynomials, allowing us to obtain numerous previously unknown A-polynomials within minutes using Mathematica.