In 1979, Hartley and Kawauchi proved that the Conway polynomial of a strongly negative amphichiral knot factors as . In this paper, we normalize the factor to define the half-Conway polynomial. First, we prove that the half-Conway polynomial satisfies an equivariant skein relation, giving the first feasible computational method, which we use to compute the half-Conway polynomial for knots with 12 or fewer crossings. This skein relation also leads to a diagrammatic interpretation of the degree-one coefficient, from which we obtain a lower bound on the equivariant unknotting number. Second, we completely characterize polynomials arising as half-Conway polynomials of knots in , answering a problem of Hartley–Kawauchi. As a special case, we construct the first examples of non-slice strongly negative amphichiral knots with determinant one, answering a question of Manolescu. The double branched covers of these knots provide potentially non-trivial torsion elements in the homology cobordism group.
Cite this article
Keegan Boyle, Wenzhao Chen, Negative amphichiral knots and the half-Conway polynomial. Rev. Mat. Iberoam. (2023), published online firstDOI 10.4171/RMI/1442