On the Kawauchi conjecture about the Conway polynomial of achiral knots

Details On the Kawauchi conjecture about the Conway polynomial of achiral knots On the Kawauchi conjecture about the Conway polynomial of achiral knots

2011-06-28

arxiv.org/abs/1106.5634v1

Mathematics

Geometric Topology

8 pages, 6 figures

Scientific paper

We give a counterexample to the 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 Kawauchi conjecture is true for quasi-arborescent knots and counterexamples in the class of alternating knots must be quasi-polyhedral.

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