Topology Proceedings 30 No. 1 (2006), pp. 153-162

Chirality and the Conway Polynomial

James Conant

In recent work with Jacob Mostovoy and Ted 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(z²) is a perfect square inside the ring of power series with integer coefficients, or, equivalently, the image of C(z)C(iz)C(z²) is a perfect square inside the ring of polynomials with Z₄ 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 Richard Hartley and Akio Kawauchi [Polynomials of amphicheiral knots, Math. Ann. 243 (1979)

Keywords: amphicheirality, Conway polynomial, Goussarov-Vassiliev invariants

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Topology Proceedings, Volume 30, No. 1 (2006)
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