[Paper Review] Chirality and the Conway polynomial

James Conant|arXiv (Cornell University)|Mar 28, 2005

Geometric and Algebraic Topology5 references3 citations

TL;DR

This paper investigates the algebraic structure of the Conway polynomial for amphicheiral knots, proposing that it factors as $f(z)f(-z)$ for some integer polynomial $f$. Using a formal logarithm and modular arithmetic, the authors show that if this factorization holds, then $C(z)C(iz)C(z^2)$ is a perfect square in $\mathbb{Z}_4[[z]]$, supporting a conjecture on Vassiliev invariants and providing evidence for the broader conjecture that all amphicheiral knots have such factorizable Conway polynomials.

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.]

Motivation & Objective

• To investigate whether the Conway polynomial of an amphicheiral knot can be written as $f(z)f(-z)$ for some integer polynomial $f$.
• To examine the consequences of this factorization on the structure of $C(z)C(iz)C(z^2)$ in the ring of power series with $\mathbb{Z}_4$ coefficients.
• To provide evidence for the conjecture that $C(z)C(iz)C(z^2)$ is a perfect square modulo 4, linking it to the integrality and torsion properties of Vassiliev invariants.
• To extend known results from strongly and negatively amphicheiral knots to the general case, using algebraic and topological techniques.

Proposed method

• Define a discrete logarithm $\operatorname{log}_{\mathbb{Z}}$ on power series with integer coefficients to construct primitive Vassiliev invariants from the Conway polynomial.
• Use the formal exponential $\operatorname{exp}_{\mathbb{Z}}$ to reverse the logarithm and reconstruct the Conway polynomial from its logarithmic coefficients.
• Apply reduction modulo 4 to analyze the structure of $C(z)C(iz)C(z^2)$, showing that if it is a square in $\mathbb{Z}_4[[z]]$, then it lifts to a square in $\mathbb{Z}[[z]]$.
• Use Proposition 4.2 to show that if a power series with integer coefficients is a square modulo 4, then its square root also has integer coefficients.
• Use Proposition 4.3 to analyze the structure of formal power series in $\mathbb{Z}_4[[z]]$ whose square is a polynomial, showing that such a series must be a polynomial modulo 2.
• Leverage known results from Hartley and Kawauchi on negative and strongly amphicheiral knots to show that their Conway polynomials factor as $f(z)f(-z)$, and thus satisfy the conjecture.

Experimental results

Research questions

• RQ1 Does the Conway polynomial of every amphicheiral knot factor as $f(z)f(-z)$ for some integer polynomial $f$?
• RQ2 If so, does this factorization imply that $C(z)C(iz)C(z^2)$ is a perfect square in $\mathbb{Z}_4[[z]]$?
• RQ3 Can the conjecture that $C(z)C(iz)C(z^2)$ is a perfect square in $\mathbb{Z}_4[[z]]$ be proven using the discrete logarithm and modular reduction techniques?
• RQ4 Is the condition $C(z) = f(z)f(-z)$ strictly stronger than the square condition in $\mathbb{Z}_4[[z]]$, and what are the implications for the determinant of the knot?
• RQ5 Are there counterexamples to the factorization $C(z) = f(z)f(-z)$ among hyperbolic amphicheiral knots with $\mathbb{Z}_{4k}$ symmetry?

Key findings

• If the Conway polynomial of an amphicheiral knot factors as $f(z)f(-z)$, then $C(z)C(iz)C(z^2)$ is a perfect square in the ring of power series with integer coefficients.
• The condition $C(z)C(iz)C(z^2)$ being a perfect square in $\mathbb{Z}_4[[z]]$ is equivalent to the existence of a polynomial $F \in \mathbb{Z}_4[[z]]$ such that $F^2 = C(z)C(iz)C(z^2)$.
• For all negative amphicheiral and strongly amphicheiral knots, the Conway polynomial factors as $f(z)f(-z)$, and thus satisfies the square condition in $\mathbb{Z}_4[[z]]$, confirming the conjecture in these cases.
• The determinant of an amphicheiral knot is a sum of two squares, a fact implied by the factorization $C(z) = f(z)f(-z)$, and this is consistent with the conjecture.
• The paper provides evidence that the conjecture $C(z) = f(z)f(-z)$ holds for all amphicheiral knots, including hyperbolic knots with $\mathbb{Z}_{4k}$ symmetry, though this remains unproven in general.
• The authors show that if $C(z)C(iz)C(z^2)$ is a square modulo 4 and its coefficients are congruent to those of a power series with integer coefficients, then the square root also has integer coefficients, supporting the lifting of the square condition from $\mathbb{Z}_4$ to $\mathbb{Z}$.