Abstract
We give a counterexample to the generalized Kawauchi conjecture on the Conway polynomial of achiral knots which asserts that the Conway polynomial C(z) of an achiral knot satisfies the splitting property C(z) = F(z)F(-z) for a polynomial F(z) with integer coefficients. We show that the Bonahon–Siebenmann decomposition of an achiral and alternating knot is reflected in the Conway polynomial. More explicitly, the generalized Kawauchi conjecture is true for quasi-arborescent knots and counterexamples in the class of alternating knots must be quasi-polyhedral.
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