# Negative amphichiral knots and the half-Conway polynomial

### Keegan Boyle

University of British Columbia, Vancouver, Canada### Wenzhao Chen

University of British Columbia, Vancouver, Canada

## Abstract

In 1979, Hartley and Kawauchi proved that the Conway polynomial of a strongly negative amphichiral knot factors as $f(z)f(−z)$. In this paper, we normalize the factor $f(z)$ 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 $S_{3}$, 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. 40 (2024), no. 2, pp. 581–622

DOI 10.4171/RMI/1442