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Conway Polynomial

The Conway polynomial del _L(x), sometimes known as the Conway-Alexander polynomial, is a modified version of the Alexander polynomial Delta_L(x) that was formulated by J. H. Conway (Livingston 1993, pp. 207-215). It is a reparametrization of the Alexander polynomial given by

Delta_L(x^2)=del _L(x-x^(-1)).

The skein relationship convention used by for the Conway polynomial is

del _(L_+)(x)-del _(L_-)(x)=xdel _(L_0)(x)

(Doll and Hoste 1991).

Examples of Alexander Delta and Conway del polynomials for common knots are given in the following table

knot KDelta_K(x)del _K(x)
trefoil knotx^(-1)-1+xx^2+1
figure eight knot-x^(-1)+3-x1-x^2
Solomon's seal knotx^(-2)-x^(-1)+1-x+x^2x^4+3x^2+1
stevedore's knot-2x^(-1)+5-2x1-2x^2
Miller Institute knot-x^(-2)+3x^(-1)-3+3x-x^2-x^4-x^2+1

For Turk's head links Th(p,q) and Th(q,p) with their closed-braid orientations, Stanggassinger (2026) proved that coefficients of powers congruent to 0 or 1 modulo 4 agree, while those congruent to 2 or 3 modulo 4 are negatives of one another.

See also

Alexander Polynomial, Turk's Head Link

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References

Doll, H. and Hoste, J. "A Tabulation of Oriented Links." Math. Comput. 57, 747-761, 1991.Livingston, C. Knot Theory. Washington, DC: Math. Assoc. Amer., 1993.Stanggassinger, W. A. "Conway Polynomials of Th(p,q) and Th(q,p): A Proof of Takemura's Conjecture." 23 Oct 2026. https://arxiv.org/abs/2610.32269.

Referenced on Wolfram|Alpha

Conway Polynomial

Cite this as:

Weisstein, Eric W. "Conway Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConwayPolynomial.html

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