Alexander ideals of classical knots

Publicacions Matem`atiques, Vol 41 (1997), 489–494. ALEXANDER IDEALS OF CLASSICAL KNOTS C. Kearton and S. M. J. Wilson Abstract The Alexander ideals of classical knots are characterised, a result which extends to certain higher dimensional knots. 1. Introduction Let Kbe the closed complement of a tubular neighbourhood of a classical knot k=S3,S1. By theorems of Hurewicz and Alexander, π1(K, ∗) abelianises to H1(K) which is isomorphic to the infinite cyclic group (t:), written multiplicatively. This yields the infinite cyclic cover ˜ K→K, and hence we obtain H1˜ Kas a finitely generated module over the ring Λ = Zt, t−1. This module has associated with it a sequence of ideals of Λ, E1⊆E2⊆E3⊆···⊆En=En+1 =···=Λ. The highest common factor of Eiis a Laurent polynomial ∆i(t), defined up to multiplication by ±tr. These are known as the ith Alexander ideal and ith Alexander polynomial of the knot. Definition 1.1. Let Aibe the set of ideals of Λ which arise as the ith Alexander ideal of some knot, and let A=∞ i=1 Ai. It is shown in Corollary 2.3 below that A1⊂A2⊆A3⊆···⊆A. The first author wishes to thank the Government of Catalunya and the Centre de Recerca Matem`atica of the Institut d’Estudis Catalans at the Universitat Aut`onoma de Barcelona for their support and hospitality. Keywords. Alexander ideal, knot. 1991 Mathematics subject classifications: 57M25, 57Q45. 490 C. Kearton, S. M. J. Wilson The first inclusion is strict, since all the ideals in A1are principal whereas those in A2are not necessarily so (see [1] for examples). A polynomial f(t)∈Λ arises as ∆i(t) for some iand some knot k if and only if f(1) = ±1 and ft−1=±trf(t) for some r(see [3] and [5]). The main result of this paper, Theorem 2.5, gives a similar characterisation of the ideals belonging to A. 2. Results For us an Alexander matrix Uof a knot will be one which presents H1˜ Kas a module over Λ. We shall assume that it is square; indeed, we could assume that U=tV −Vwhere Vis a Seifert matrix of the knot. Definition 2.1. Let Ube an m×nmatrix with entries in Λ. For each integer k≥1, the kth elementary ideal Ek(U) is defined as follows. (1) If 0 ≤n−k<m, then Ek(U) is the ideal generated by the (n−k+1)×(n−k+ 1) minors of U. (2) If n−k≥m, then Ek(U) = (0). (3) If n−k<0, then Ek(U)=Λ. Note that this differs slightly from the definition of elementary ideal given in [1], because our Alexander matrices are square. As shown in [1], the elementary ideals of Uform an ascending chain E1(U)⊆E2(U)⊆···⊆En(U)=En+1 (U)=···=Λ. If Uis an Alexander matrix of a knot then, of course, the sequence of elementary ideals of Udepends only on the knot, and is the sequence of Alexander ideals. Lemma 2.2. Let Ube an n×nmatrix with entries in Λ. Then for every integer r>0there is an (n+r)×(n+r)matrix Vsuch that the sequence E1+r(V)⊆E2+r(V)⊆···⊆En+r(V) is the same as E1(U)⊆E2(U)⊆···⊆En(U). Proof: Let ∆ = det U, and first consider the case r= 1. Let V=U0 0∆  Alexander ideals 491 so that Vis an (n+1)×(n+ 1) matrix. For 1 ≤s≤n,Es+1 (V)is generated by the (n−s+1)×(n−s+ 1) minors of V, and the only possible non-zero ones are the (n−s+1)×(n−s+ 1) minors of U and ones of the form ∆ times an (n−s)×(n−s) minor of U. Since ∆∈(∆) = E1(U)⊆Es(U), these generate Es(U). This establishes the result for r= 1; repeated application of the argument gives the general case. Corollary 2.3. A1⊂A2⊆A3⊆···⊆A. Proof: If Uis an Alexander matrix, then (det U) is also an Alexander matrix, hence so is Vin the proof above. Define the map ε:Λ→Zto be the ring homomorphism sending t→ 1, and define conjugation in Λ to be the ring homomorphism sending t→ t−1, denoted t.IfUis the Alexander matrix of a classical knot, then the following properties of Ei(U) are well known (see [1]). Ei(U)=Ei(U), ε(Ei(U)) = Z for all i≥1. Lemma 2.4. Let g∈Λsatisfy g(1) = 1. Then there is a classical knot kwhich has H1˜ K∼ =Λ (g)⊕Λ (g). Proof: Let Mbe the module above, generated by u, v such that gu = 0=gv. Define an hermitian pairing (,):M×M→Λo/Λ, where Λois the field of fractions of Λ, by (u, v)=1 g=(v,u), (u, u)=0=(v,v). The map (1 −t):M→Mis an isomorphism. In fact, (g(1) −g(t)) u=g(1)u−g(t)u=u 492 C. Kearton, S. M. J. Wilson and g(1) −g(t)=(1−t)a(t) for some a(t)∈Λ. A similar argument holds for v,g; hence the map is onto. Suppose that (1 −t)w= 0. Then tw =w, and so g(t)w=g(1)w=w, g(t)w=g(1)w=w. But g(t)g(t) annihilates M,so0=g(t)g(t)w=g(t)w=w, and hence the map is one-one. The pairing defined above is nonsingular, in the sense that the adjoint map θ:M→Hom (M,Λo/Λ) is an isomorphism. To see this, suppose that θ(w) = 0, where w= au +bv. Then 0=(au +bv, u)= b g∈Λo/Λ and so b=βgfor some β∈Λ. Thus w=au+bv =au+βgv =au.Now 0=(au, v)=a g∈Λo/Λ and so a=αg for some α∈Λ. Thus w=au =αgu = 0. Hence θis a monomorphism. Let f∈Hom (M,Λo/Λ), and let a, b ∈Λ satisfy f(u)=a g,f(v)= b g. Then (bu +av, u)=a(v,u)=a g=f(u) (bu +av, v)=b(u, v)= b g=f(v) and so θ(bu +av)=f. It is shown in [2] (or [4], or [6]) that any finitely generated Λ-torsionmodule Mfor which multiplication by 1−tis an isomorphism and which supports a nonsingular hermitian pairing arises as H1˜ Kfor some classical knot. Alexander ideals 493 Theorem 2.5. Let Ebe an ideal of Λ. Then E∈Aif and only if E satisfies E=Eand ε(E)=Z. Proof: As remarked above, the necessity is well known. So suppose that Esatisfies the two conditions. By the Hilbert Basis Theorem, the ideal E∩Z[t]ofZ[t] is finitely generated over Z[t]. A set of generators of E∩Z[t] over Z[t] is clearly a set of generators of Eover Zt, t−1,so Eis finitely generated, say E=(f1(t),... ,f n(t)). We claim that Ehas a set of generators g1(t),... ,g n(t) with g1(1) = ··· =gn(1) = 1. For not all of the fi(1) can be zero, since ε(E)=Z. Multiplying by −1 if necessary, we can renumber so that f1(1) ≥f2(1) ≥f3(1) ≥···≥fn(1) ≥0. If fn(1) = 0, then replace this set of generators by f1(t),... ,f n−1(t), f1(t)+fn(t). Repeat this process of replacement and renumbering until we have a set of generators f1(t),... ,f n(t) with f1(1) ≥f2(1) ≥f3(1) ≥···≥fn(1) >0. If they all evaluate to 1, well and good. If not, we must have f1(1) > fn(1), since ε(E)=Z. Replace f1(t)byf1(t)−fn(t). If the new set of generators is denoted by f 1(t),... ,f n(t), then f i(1) >0 for all iand n≤ n  i=1 f i(1) < n  i=1 fi(1). Continuing in this way we arrive at the desired set of generators g1(t),... ,g n(t). For each gi, let kibe the knot whose existence is guaranteed by Lemma 2.4, and let k=k1+···+kn. Then H1˜ Kis the orthogonal direct sum of the H1˜ Ki, and has a diagonal Alexander matrix U with entries g1, g1,... ,g n, gnon the diagonal. Clearly E2n(U)=E. Remark. Using the results of [2] (or [4], or [6]), we see that the set of Alexander ideals is the same for each set of simple (4q+ 1)-knots, q=0,1,2,..., since the set of modules and pairings which arise is the same in each dimension. This extends easily to the simple (4q−1)- knots, q=1,2,..., by defining a skew-hermitian pairing instead of an hermitian pairing in the proof of Lemma 2.4, and noting that the form of the pairing ensures that the signature of the associated quadratic form is zero (this is needed for the case q= 1, where the signature must be a multiple of 16). 494 C. Kearton, S. M. J. Wilson 3. Questions As mentioned in the Introduction, it is known that A1=A2. Is it true that An=An+1 for all n? For finitely many n? Only for n= 1? Can one characterise An? And what of the sequence of ideals? Can any sequence be realised, subject to each Eisatisfying the two conditions known to be necessary? The corresponding question for Alexander polynomials has been answered in the affirmative by Levine in [3]. References 1. R. H. Crowell and R. H. Fox,“Introduction to Knot Theory,” Springer-Verlag, New York, 1963. 2. C. Kearton, Classification of simple knots by Blanchfield duality, Bull. Amer. Math. Soc. 79 (1973), 952–955. 3. J. Levine, A characterization of knot polynomials, Topology 4 (1965), 135–141. 4. J. Levine, Knot modules, Trans. Amer. Math. Soc. 229 (1977), 1–50. 5. H. Seifert,¨ Uber das Geschlect von Knoten, Math. Ann. 110 (1934), 571–592. 6. H. F. Trotter, Knot modules and Seifert matrices, in “Knot Theory, Proceedings, Plans-sur-Bex 1977,” Lecture Notes in Math. 685, Springer-Verlag, New York, 1978, pp. 291–299. Department of Mathematics Durham University South Road Durham, DH1 3LE ENGLAND e-mail: Cherry[email protected] e-mail: [email protected] Primera versi´o rebuda el 6 de Maig de 1996, darrera versi´o rebuda el 3 de Setembre de 1996