On reciprocality of twisted Alexander invariants
Jonathan A. Hillman (University of Sydney), Daniel S. Silver, (University of South Alabama), Susan G. Williams (University of South, Alabama)
TL;DR
This paper investigates the reciprocality of twisted Alexander invariants for knots, establishing conditions under which the associated Reidemeister torsion is reciprocal, and providing a counterexample when these conditions are not met.
Contribution
It characterizes when twisted Alexander invariants are reciprocal for SL(n,C) representations and presents a counterexample in SL(3,Z) showing non-reciprocity.
Findings
Reciprocity holds when the representation is conjugate to its dual.
A specific SL(3,Z) representation demonstrates non-reciprocity.
Conditions for reciprocality depend on the conjugacy to the dual representation.
Abstract
Given a knot and an SL(n,C) representation of its group that is conjugate to its dual, the representation that replaces each matrix with its inverse-transpose, the associated twisted Reidemeister torsion is reciprocal. An example is given of a knot group and SL(3,Z) representation that is not conjugate to its dual for which the twisted Reidemeister torsion is not reciprocal.
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