Uniform representations and character varieties of odd Turk's head knots

Abstract

Let τ be the reflection reversing the order of the punctures in a p-punctured disc, with p odd. For any braid b such that K = closure(b(τbτ)) is a knot with det K > 1, I construct an SU(2) representation of the non-orientable mapping torus of bτ. Its restrictions to the fibre, the orientable double cover, and both boundary Klein bottles are irreducible. The construction is uniform: an integral anti-symplectic operator for the half-monodromy provides a nonconstant fixed colouring over a finite field of odd characteristic. I then apply Choi–Porti–Yoon's Theorem 3.5 to obtain dim_C X(Th(p,q)) ≥ 2 for odd coprime p,q ≥ 3. The dimension estimate is an explicitly cited external dependency.