Kauffman's polynomial and alternating links

Abstract

These theorems result in quick tests which will often distinguish between links having alternating diagrams with the same number of crossings; links having reduced alternating diagrams with different numbers of crossings are automatically distinguished by Corollary 1 of [12] (proved also in [4] and [9]). Also, as L. H. Kauffman has pointed out, it follows from Theorem 1 that any reduced alternating diagram of an amphicheiral alternating knot must have writhe 0. K. Murasugi, in [lo], has independently obtained a different proof of Theorem 1; he has discovered a formula relating the extreme powers of t in the Jones polynomial VL(t) with the writhe of the reduced alternating diagram of L and the signature of L. I am grateful to W. B. R. Lickorish for making the following important observation: since F,(u, z) determines VL(t) (see [7]), the combination of these two proofs of Theorem 1 yields the result that, for alternating links L, F,(a, z) determines the signature. That the polynomial F,(a, z) does not determine the signature in general is evidenced by the knot 9,, and its obverse: these knots share the same Kauffman polynomial, yet have different signatures. Theorem 2 may be regarded as a tentative first step towards settling the famous “flyping conjecture” (see [ 111, $2). In proving the results of this article, substantial use is made of the ideas in M. E. Kidwell’s paper [6]. We use the same terminology as in [ 121, except that in deference to custom we now say that a diagram with no nugatory crossing m is reduced (rather than “irreducible”). As in [12], a diagram is prime if it is not decomposable as a non-trivial diagrammatic connected sum .Of course, a prime diagram might represent a composite link, or indeed a split link; moreover, a connected, non-prime diagram can represent a prime link. The reader is also referred to [12] for graph-theoretical definitions and background information on the Tutte polynomial. The original paper introducing the