Characteristic Properties of the Knot Polynomials

Abstract

A survey of the knot polynomials Δ _k (t) computed at the end of the preceding chapter shows that, for each of them, Δ _k (1) = ± 1. A proof that this equation holds for all knot polynomials is the objective of the first section of the present chapter. The survey also substantiates the assertion that all knot polynomials are reciprocal polynomials, i.e., for every knot polynomial Δ _k (t), there exists an integer n such that Δ _k (t) = t ⁿ Δ _k (t ^-1). Thus, if Δ _k (t) = c _n t ⁿ + c _n-1 t ^n-1 +··· + c ₀, the coefficients exhibit the symmetry c _i = c _n-i , i = 0, ··· , n. As was pointed out in Section 3 of Chapter VIII, this property is essential to our conclusion that knots of the same type possess identical polynomials. It is therefore important to close this gap in the theory. The proof that knot polynomials are reciprocal polynomials will be effected in Sections 2 and 3 by introducing the notion of dual group presentations, the crucial examples of which are the over and under presentations of knot groups defined in Chapter VI. It should be emphasized that our arguments apply only to tame knots, and throughout this chapter “knot” always means “tame knot.”