Some knots with no SU(2)-abelian surgeries

Abstract

A knot in $S^3$ is said to be \emph{not} $SU(2)$-abelian knot, if every non-trivial surgery along it yields a $3$-manifold whose fundamental group admits an irreducible $SU(2)$-representation. We provide examples of knots in $S^3$ that are not $SU(2)$-abelian. A knot $K$ is said to be $SU(2)$-\emph{clean} if whenever the fundamental group of the integer $r$-surgery $\fund{K(r)}$ has no $SU(2)$-irreducible representation, then the Alexander polynomial of $K$ does not vanish at any $r$-th root of unity. We show that if $K$ is a non-trivial $SU(2)$-clean knot, the connected sum $K\#K$ is never $SU(2)$-abelian. Finally, combining known results on $SU(2)$-abundant knots and a classical epimorphism between knot groups, we show that \emph{every} non-torus knot with at most $9$ crossings is not $SU(2)$-abelian, with at most the two exceptions of $9_{47}$ and $9_{49}$.