Chirality and the Conway polynomial

Abstract

In recent work with J.Mostovoy and T.Stanford,the author found that for every natural number n, a certain polynomial in the coefficients of the Conway polynomial is a primitive integer-valued degree n Vassiliev invariant, but that modulo 2, it becomes degree n-1. The conjecture then naturally suggests itself that these primitive invariants are congruent to integer-valued degree n-1 invariants. In this note, the consequences of this conjecture are explored. Under an additional assumption, it is shown that this conjecture implies that the Conway polynomial of an amphicheiral knot has the property that C(z)C(iz)C(z^2) is a perfect square inside the ring of power series with integer coefficients, or, equivalently, the image of C(z)C(iz)C(z^2) is a perfect square inside the ring of polynomials with Z_4 coefficients. In fact, it is probably the case that the Conway polynomial of an amphicheiral knot always can be written as f(z)f(-z) for some polynomial f(z) with integer coefficients, and this actually implies the above "perfect squares" conditions. Indeed, by work of Kawauchi and Hartley, this is known for all negative amphicheiral knots and for all strongly positive amphicheiral knots. In general it remains unsolved, and this paper can be seen as some evidence that it is indeed true in general. [Added 2/22/12: Of note is the recent paper arXiv:1106.5634v1 [math.GT] by Ermotti, Hongler and Weber, which finds a counterexample to the conjecture that all Conway polynomials of amphicheiral knots are of the form f(z)f(-z). Intriguingly, their main example still satisfies C(z)C(iz)C(z^2)=f(z)^2 for a power series f(z), making the main conjecture of the present paper that much more compelling, in the author's opinion.]