Alexander polynomials of non-locally-flat knots Greg Friedman Yale University Dept. of Mathematics 10 Hillhouse Ave PO Box 208283 New Haven, CT 06520 [email protected] Tel. 203-432-6473 Fax: 203-432-7316 October 1, 2002 Abstract We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties. For knots with point singularities, we obtain a classification of these polynomials that is complete except for one special low-dimensional case. This classification extends existing classifications for PL locally-flat knots. For knots with higher-dimensional singularities, we further extend the necessary conditions on the invariants. We also construct several varieties of singular knots to demonstrate realizability of certain families of polynomials as generalized Alexander polynomials. These constructions, of independent interest, generalize known knot constructions such as frame spinning and twist spinning. 1 Contents 1 Introduction 2 2 Polynomial algebra 8 3 Sphere knots with point singularities and locally-flat diskknots 11 3.1 Introduction...................................... ........ 11 3.2 TheKnotComplement................................. ....... 11 arXiv:math/0307328v1 [math.GT] 24 Jul 2003 3.3 Necessary conditions on the Alexander invariants . .............. 12 3.3.1 Alexander invariants . ..... 12 3.3.2 Constructionofthecovering . ....... 15 3.3.3 LinkingNumbers.................................... ... 17 3.3.4 TheproofofTheorem3.1 ............................. ..... 18 3.4 Some corollaries; Definition of Alexander polynomials . ........... 19 3.5 Realization of given polynomials . ....... 21 2 4 3.5.1 Realizing λi for D ⊂ D .................................. 21 n−2 n 3.5.2 Realizing λi for D ⊂ D , n ≥ 5............................. 22 3.5.3 Realization of all Alexander polynomials . ...... 28 3.6 The middle dimension polynomial . ...... 30 3.6.1 The Blanchfield pairing . ... 31 12000 Mathematics Subject Classification: Primary 57Q45; Secondary 57Q35, 57N80 1 3.6.2 Realization of middle dimensional pairings . ...... 32 3.6.3 Matrix representations of the middle dimension module and its pairing......... 33 3.6.4 Characterization of the middle dimension polynomial in terms of pairings . 38 3.7 Conclusion ......................................... ..... 40 4 Knots with more general singularities 40 4.1 Introduction...................................... ........ 40 4.2 Necessary conditions on the Alexander invariants . .............. 41 4.2.1 Geometric preliminaries . .... 41 4.2.2 Duality and normalization theorem . ...... 42 4.2.3 Polynomials of the boundary “knot” . ....... 48 4.2.4 The subpolynomials . ... 55 4.2.5 High dimensions . .. 56 4.3 Constructions..................................... ........ 57 4.3.1 Framespinning..................................... ... 57 4.3.2 Frame twist-spinning . ..... 62 4.3.3 Suspensions...................................... .... 68 1 Introduction Background. One of the central motivations for studying knots and their invariants, including Alexander polynomials, is the central role that knots play in the understanding of the geometry of subvarieties of real codimension two. Thus, for a subpseudomanifold, Xn−2, of a manifold W n, e.g. for a complex divisor of a complex manifold, the local geometry of X in W is classically described in terms of link pairs. Therefore, there is a large classical topological and algebraic geometric literature (e.g. [27], [22], [23], [24]) which studies the topology of the non-singular knots that arise as the link pairs of isolated singular points. However, in general, the singularities that arise naturally in (both high- and low-dimensional) topological and algebraic geometric situations (see, e.g., [5], [6], [12]) cannot be assumed to be isolated, and the corresponding link pairs of points of the singularities will consist of knotted sphere pairs which are themselves singular embeddings. In the non-singular case, more specifically, the focus of knot theory historically has been the study of smooth or locally-flat codimension two knots, that is embeddings of Sn−2 in Sn which are differentiable or piecewise-linear such that the neighborhood pair of any image point is PL-homeomorphic to an unknotted ball pair Dn−2 ⊂ Dn. Furthermore, much effort has gone into the study of invariants of knots, algebraic objects which can be assigned to knots and which are identical for equivalent knots. Prominent among these invariants are the Alexander polynomials which are elements, up to similarity class, of the ring of integral Laurent polynomials Λ := Z[Z] =∼ Z[t,t−1]. They can be defined in may ways, one of which is as follows: By Alexander duality, the knot complement C := Sn −Sn−2 is a homology circle and hence possesses an infinite cyclic cover, C˜. The homology of C˜ with rational coefficients, Hi(C˜; Q), has the structure of a module over Γ := Q[t,t−1], where the action of t is given by the covering translation. These modules can be shown to be Γ-torsion modules which, since Γ is a principal ideal domain, possess square presentation matrices. The determinants of these matrices are elements of Γ which can be “normalized” by “clearing denominators” to elements of Λ whose coefficients are relatively prime, collectively. These normalized determinants are the Alexander polynomials. An equivalent approach would be to begin with the homology modules of C taken with a local coefficient system Γ , which is given by stalk Γ and action determined by factoring the fundamental group to the group of covering translations. In [19], Levine completely characterized the Alexander polynomials of PL-locally-flat knots. (Some of these results were known somewhat earlier for low dimensions; see [7], [32], [17], [15].) If we represent the polynomial corresponding to the homology group in dimension i by pi, he showed that the following conditions are necessary and sufficient for the collection {pi},0 <i<n−1, to be the Alexander polynomials of such a knot Sn−2 ⊂ Sn (in the other dimensions the polynomials are trivial): −1 1. pi(t) ∼ pn−i−1(t ) (“∼” denotes similarity in Λ), 2 2. pi(1) = ±1, 3. if n =2q + 1, q even, then p(−1) is an odd square. In this paper, we study the generalization of these invariants and their properties to various classes of knots which are not necessarily locally-flat, that is non-locally-flat (we also sometimes refer to knots which are definitely not locally-flat as singular). For knots with only point singularities, we establish necessary and sufficient conditions generalizing those of Levine and which form a complete characterization in all dimensions save n = 5 (and even for n = 5, we come close to a complete characterization; see below). For knots with more general singularities, we further generalize the necessary conditions, and we study several methods for realizing given sets of polynomials, including a construction of independent interest in the study of smooth knots which generalizes twist spinning [40], superspinning [4], and frame spinning [30]. We now outline our results section by section. Section 2: Polynomial algebra. This is a preliminary section in which we develop some fundamental results of what we call polynomial algebra by analogy with homological algebra. In particular, to each torsion Γ-module there is associated an element of Γ, up to similarity class. This is the determinant of the presentation matrix of the module or, equivalently, the product of its torsion coefficients (recall that Γ = Q[Z] = Q[t,t−1] is a principal ideal domain). We develop some relationships among the polynomials associated to the modules of an exact sequence. Section 3: Sphere knots with point singularities and locally-flat disk knots. In this section, we first show that the study of the complement of a knot with a point singularity is homologically equivalent to the study of the complement of a locally-flat proper disk knot whose boundary sphere knot is the link pair of the singular point. In fact, by a technique of Milnor and Fox [26], the same is true of a sphere knot with any finite number of point singularities, and the boundary sphere knot will be the knot sum of the link pair knots of all the singular points. In this context, we define a family of three polynomials: λi, corresponding to the homology module of the cover of the disk knot complement, C; νi, corresponding to the boundary sphere knot complement, X; and µi, corresponding to the relative homology of the cover of the pair (C,X). Furthermore, there is a natural factorization of these polynomials: νi ∼ aibi, λi ∼ bici, and µi ∼ ciai−1. With this notation we prove the following theorem: Theorem 1.1 (Theorem 3.28). For n 6= 5 and 0 <i<n − 1, 0 <j<n − 2, the following conditions are necessary and sufficient for λi, µi, and νj to be the polynomials associated to the Γ-modules Hi(C˜; Q), n−2 n Hi(C,˜ X˜; Q), and Hj (X˜; Q) of a locally-flat proper disk knot D ⊂ D : There exist polynomials ai(t), bi(t), and ci(t), primitive in Λ, such that 1. (a) νi ∼ aibi (b) λi ∼ bici (c) µi ∼ ciai−1 −1 2. (a) ci(t) ∼ cn−i−1(t ) −1 (b) ai(t) ∼ bn−i−2(t ) 3. ai(1) = ±1,bi(1) = ±1,ci(1) = ±1, a0(t)=1. ω 4. If n = 2q +1 and q is even, then there exist an integer ρ and an integer ω ≥ 0 such that (1−t) ρ ±cq (t) is the discriminant of a skew Hermitian form A × A → Q(Λ)/Λ on a finitely-generated Λ-module, A, on which multiplication by t − 1 is an isomorphism (or equivalently, cq(t) =det[M(t)], where M(t) = (−1)q+1(R−1)′τRt − τ ′ for integer matrices τ and R such that R has non-zero determinant and (R−1)′τR is an integer matrix (here ′ indicates transpose); see Section 3.6 for more details).