For twisted Alexander vanishing groups, every faithful irreducible representation of the knot group makes the twisted Alexander polynomial vanish, with explicit knot constructions provided.
Turaev, Introduction to Combinatorial Torsions , Birkh¨ auser, Basel (2001)
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Twisted Alexander vanishing groups of knots
For twisted Alexander vanishing groups, every faithful irreducible representation of the knot group makes the twisted Alexander polynomial vanish, with explicit knot constructions provided.