On Knots with Trivial Alexander Polynomial

We use the 2-loop term of the Kontsevich integral to show that there are (many) knots with trivial Alexander polynomial which don’t have a Seifert surface whose genus equals the rank of the Seifert form. This is one of the first applications of the Kontsevich integral to intrinsically 3-dimensional questions in topology. Our examples contradict a lemma of Mike Freedman, and we explain what went wrong in his argument and why the mistake is irrelevant for topological knot concordance. 1. A question about classical knots Our starting point is a wrong lemma of Mike Freedman in [F, Lemma 2], dating back before his proof of the 4-dimensional topological Poincaré conjecture. To formulate the question, we need the following Definition 1.1. A knot in 3-space has minimal Seifert rank if it has a Seifert surface whose genus equals the rank of the Seifert form. Since the Seifert form minus its transpose gives the (nonsingular) intersection form on the Seifert surface, it follows that the genus is indeed the smallest possible rank of a Seifert form. The formula which computes the Alexander polynomial in terms of the Seifert form shows that knots with minimal Seifert rank have trivial Alexander polynomial. Freedman’s wrong lemma claims that the converse is also true. However, in the argument he overlooks the problem that S-equivalence does not preserve the condition of minimal Seifert rank. It turns out that not just the argument, but also the statement of the lemma is wrong. This has been overlooked for more than 20 years, maybe because none of the classical knot invariants can distinguish the subtle difference between trivial Alexander polynomial and minimal Seifert rank. In the last decade, knot theory was overwhelmed by a plethora of new “quantum”invariants, most notably the HOMFLY polynomial (specializing to the Alexander and the Jones polynomials), and the Kontsevich integral. Despite their rich structure, it is not clear how strong these invariants are for solving open problems in low dimensional topology. It is the purpose of this paper to provide one such application. Theorem 1. There are knots with trivial Alexander polynomial which don’t have minimal Seifert rank. More precisely, the 2-loop part of the Kontsevich integral induces an epimorphism Q from the monoid of knots with trivial Alexander polynomial, onto an infinitely generated abelian group, such that Q vanishes on knots with minimal Seifert rank. The easiest counterexample is shown in Figure 1. We use clasper calculus to draw the example, which is explained in Section 5. This geometric calculus is a wonderful tool to organize knots. In Figure 1, the clasper amplifies the relevant features of the example. Moreover, if one pulls the central edge of the clasper out of the visible Seifert surface, one obtains an S-equivalence to a nontrivial knot with minimal Seifert rank. Remark 1.2. All of the above notions make sense for knots in homology spheres. Our proof of Theorem 1 works in that setting, too. Since [F, Lemma 2] was the starting point of what eventually became Freedman’s theorem that all knots with trivial Alexander polynomial are topologically slice, we should make sure that the above counterexamples to his lemma don’t cause any problems in this important theorem. Fortunately, an argument independent Date: This edition: May 31, 2002 First edition: May 31, 2002. The authors are partially supported by NSF grants DMS-02-03129 and DMS-00-72775 respectively. This and related preprints can also be obtained at http://www.math.gatech.edu/∼stavros and http://math.ucsd.edu/∼teichner 1991 Mathematics Classification. Primary 57N10. Secondary 57M25.