Colorability and determinants of $T(m,n,r,s)$ twisted torus knots for $n\equiv\pm 1\pmod m$

Abstract

We develop theorems to compute the $p$-colorability of the families of $T\left(m,n,r,s\right)$ twisted torus knots for $n\equiv \pm 1\left(modm\right)$ by finding their determinants. Instead of the usual method of reducing crossing matrices to find the determinant, we describe a new method that is applicable for braid representations with full cycles and twists.