On the twisted Alexander polynomial for representations into SL_2(C)

We study the twisted Alexander polynomial ${\mathrm{\Delta }}_{K,\rho }$ of a knot $K$ associated to a non-abelian representation $\rho$ of the knot group into $S{L}_2(\text{BC})$. It is known for every knot $K$ that if $K$ is fibered, then for every non-abelian representation, ${\mathrm{\Delta }}_{K,\rho }$ is monic and has degree $4g(K)-2$ where $g(K)$ is the genus of $K$. Kim and Morifuji recently proved the converse for 2-bridge knots. In fact they proved a stronger result: if a 2-bridge knot $K$ is non-fibered, then all but finitely many non-abelian representations on some component have ${\mathrm{\Delta }}_{K,\rho }$ non-monic and degree $4g(K)-2$. In this paper, we consider two special families of non-fibered 2-bridge knots including twist knots. For these families, we calculate the number of non-abelian representations where ${\mathrm{\Delta }}_{K,\rho }$ is monic and calculate the number of non-abelian representations where the degree of ${\mathrm{\Delta }}_{K,\rho }$ is less than $4g(K)-2$.