Negative amphichiral knots and the half-Conway polynomial

TL;DR

This paper introduces the half-Conway polynomial for negative amphichiral knots, providing a computational method, characterizing possible polynomials, and constructing examples with implications for knot theory and homology cobordism.

Contribution

It defines the half-Conway polynomial, establishes an equivariant skein relation for computation, characterizes all such polynomials, and constructs new non-slice knots with specific properties.

Findings

Developed a computational method for half-Conway polynomial

Characterized all polynomials that can arise as half-Conway polynomials

Constructed non-slice strongly negative amphichiral knots with determinant one

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.