Polynomials of equivariantly slice knots

Jonathan A. Hillman

School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, AUSTRALIA

e-mail: jonh@maths.usyd.edu.au

ABSTRACT

We show that if K is a knot with semifree period q and which is equivariantly slice then the Murasugi polynomial Δ_{Z/qZ}(K) may be expressed as a quotient aā/q^{2n}, for some a ∈ ℤ[t, t^{−1}, u]/(u^{q} − 1) and n ≥ 0.

AMS Subject Classification (1991): Primary 57M25. Key words and phrases: Alexander polynomial. concordance. equivariant. knot. Murasugi polynomial. ribbon. slice.

If K is a knot invariant under a rotation h of order q about a disjoint axis A let Δ_{Z/qZ}(K) be the image of Δ_{1}(\overline{A} ∪ \overline{K}) in the ring ℤ[Z × Z/qZ] = ℤ[t, t^{−1}, u]/(u^{q} − 1), where \overline{A} ∪ \overline{K} is the orbit link in S^{3}/⟨h⟩ and u, t are the meridians corresponding to \overline{A} and \overline{K}, respectively. Davis and Naik used Milnor-Reidemeister torsion to show that there are a = a(t, u) and b = b(t, u) in ℤ[t, t^{−1}, u]/(u^{q} − 1) such that a(1, u) = b(1, u) = 1 and Δ_{Z/qZ}(K)b\overline{b} = a\overline{a}. Moreover if a(t, u) ∈ ℤ[t, t^{−1}, u]/(u^{q} −1) and a(1, u) = 1 then aā = Δ_{Z/qZ}(K) for some equivariantly ribbon knot K with semifree period q [DN02]. This suggests a possible means of distinguishing between equivariantly ribbon and equivariantly slice knots (although there are as yet no candidates to test).

In Theorem 8.22 of [Hi] we attempted to recover the Davis-Naik result by means of Blanchfield duality with rational coefficients, and found instead that Δ_{Z/qZ}(K) = a\overline{a}/d^{2}, for some a = a(t, u) ∈ ℤ[t, t^{−1}, u]/(u^{q} − 1) such that a(1, 1) = d ≠ 0. Here we shall show that we may always assume that the denominator d is a power of q. (It remains an open question whether in fact we may assume d = 1). Instead of using rational coefficients, we shall localize in two ways: (a) we invert q, so that ℤ[\frac{1}{q}][Z/qZ] is a direct sum of Dedekind domains; and (b) we invert t−1, so that we may reduce to modules which are finitely generated over ℤ[\frac{1}{q}]. (A similar reduction is useful in relating Seifert forms to Blanchfield pairings).

Let Γ = ℤ[u]/(u^{q} − 1) = ℤ[Z/qZ], Λ = ℤ[t, t^{−1}], Λ_{2} = ℤ[t, t^{−1}, u, u^{−1}], and Ψ = Λ⊗_{ℤ}Γ = Λ_{2}/(u^{q} −1) = ℤ[Z ×Z/qZ]. Let Γ^{+} = Γ[\frac{1}{q}] = ℤ[\frac{1}{q}, u]/(u^{q} −1) = ℤ[\frac{1}{q}][Z/qZ], Λ^{+} = Λ[\frac{1}{q}, (t − 1)^{−1}] and Ψ^{+} = Λ^{+} ⊗_{ℤ} Γ^{+}. If M is a Γ-module let M^{+} = Γ^{+} ⊗_{Γ} M. We shall use an overbar to denote both the canonical involution of the group ring Ψ, and also the conjugate module \overline{M} derived from M via this involution. (In other respects our notation shall be based on that of Chapter 8 of [Hi]). Let z = (t − 1)^{−1}. Then t = z^{−1}(z − 1) and Λ^{+} = ℤ[\frac{1}{q}, z, z^{−1}, (z + 1)^{−1}].

Fix a primitive q^{th} root of unity ζ_{q} and let χ(m + qZ) = ζ_{q}^{m} for m ∈ Z. Let e_{j} = \frac{1}{q}Σ_{\{g∈Z/qZ\}}χ(g)^{j}g, for each 1 ≤ j ≤ q. Then \{e_{j} | 1 ≤ j ≤ q\} is the set of canonical idempotents in ℂ[Z/qZ]. For each divisor d of q let ζ_{d} = ζ_{q}^{q/d} and I(d) = {i | 1 ≤ i ≤ n/d, (i, n/d) = 1}. Then E_{d} = Σ_{i∈I(n/d)}e_{in/d} is the sum of the Galois conjugates of e_{n/d}, and {E_{d}}_{d|q} is the set of canonical idempotents in Γ^{+}. Moreover E_{d}Γ^{+} ≅ Γ^{+}/(ϕ_{d}(u)) = ℤ[\frac{1}{q}, ζ_{d}], where ϕ_{d} is the d^{th} cyclotomic polynomial. Hence Γ^{+} ≅ ⊕_{d|q}E_{d}Γ^{+} is a direct sum of Dedekind domains. Note that the extension from Γ to Γ^{+} is flat, whereas that from Γ to ⊕_{d|q}Γ/(ϕ_{d}(u)) is not. Lemma 1. Let R be a ring and M a finitely generated R[Z]-module such that δM = 0 for some monic polynomial δ ∈ R[Z]. Then M is finitely generated as an R-module. Proof. If {m_{1}, …, m_{r}} generates M over R[Z] and δ has degree n then {Z^{i}m_{j} | 0 ≤ i < n, 1 ≤ j ≤ r} generates M over R. □ Lemma 2. Let P be a finitely generated Ψ-module such that P^{+} is annihilated by a monic polynomial. Then P^{+} = Ψ^{+} ⊗_{ℤ[ \frac{1}{q} ,z]} P̃, where P̃ is an ℤ[\frac{1}{q}, z, u]/(u^{q} − 1)-submodule which is finitely generated as a ℤ[\frac{1}{q}]-module. Proof. We may take P̃ to be the ℤ[\frac{1}{q}, z, u]/(u^{q} − 1)-submodule generated by a finite subset of P which generates P as a Ψ-module. □ Lemma 3. Let P̃ be a Γ^{+}-module which is finitely generated as a ℤ[\frac{1}{q}]-module and ℤ-torsion free. Then P̃ is projective as a Γ^{+}-module. Proof. The orthogonal idempotents {e_{d}}_{d|q} determine a direct sum decomposition P̃ ≅ ⊕e_{d}P̃. As each summand e_{d}P̃ is finitely generated and ℤ-torsion free, it is projective as a Γ^{+}e_{d}-module and hence P̃ is projective as a Γ^{+}- module. □

Theorem. Let K be a 1-knot with semifree period q about an axis A and which is equivariantly slice. Then |lk(A, K)| = 1 and we may normalize our choice of Δ_{1}(\overline{A} ∪ \overline{K}) so that Δ_{Z/qZ}(K) = a\overline{a}/q^{2m} for some a = a(t, u) ∈ ℤ[t, t^{−1}, u]/(u^{q} − 1) such that a(1, u) = q^{m}. Proof. The first assertion follows on considering the fixed point set of the restriction of the action to an invariant slice disc. Hence we may use the Torres conditions (Corollary 5.3.1 of [Hi]) to normalize Δ_{Z/qZ}(K) to be fixed under conjugation and augment to 1.

Let X = X(K), Y = X(A ∪ K) and \overline{Y} = X(\overline{A} ∪ \overline{K}) be the exteriors of the knot K and the links A ∪ K and \overline{A} ∪ \overline{K}, and let X′ and Y′ = \overline{Y}′ be their universal covering spaces. Let Y^{γ} be the preimage of Y in X′. Then meridians for A lift to loops in Y^{γ}.

A meridian for A determines an element m_{A} ∈ H_{1}(Y^{γ}; ℤ), such that (t − 1)m_{A} = (u − 1)m_{A} = 0, and M = H_{1}(X; Λ) = H_{1}(Y^{γ}; ℤ)/⟨m_{A}⟩, by excision. Then M is a finitely generated Ψ-module, and M = (t − 1)M and is ℤ-torsion free. In particular, M^{+} ≅ ℤ[\frac{1}{q}] ⊗ M. If we rewrite the Alexander polynomial Δ_{K}(t) = Δ_{0}(M) in terms of t = z^{−1}(z + 1) and clear denominators the resulting polynomial in z is monic, and so Lemma 1 applies to M^{+} and its submodules.

Let H = H_{1}(\overline{Y}; Λ_{2}) = H_{1}(Y′; ℤ). Then H has a square presentation matrix over Λ_{2} (see §7.1 of [Hi]), and the Wang sequence for the projection of Y′ on Y^{γ} gives a short exact sequence

0 → H/(u^{q} − 1)H → H_{1}(Y^{γ}; ℤ) → Z → 0,

which is split by the inclusion of m_{A}, since |lk(A, L)| = 1. Then M^{+} ≅ Ψ^{+} ⊗_{Λ_{2}} H, and so M^{+} has a square presentation matrix over Ψ^{+}. Hence pd_{Ψ^{+}}M^{+} ≤ 1 and E_{0}(M^{+}) = (Δ_{Z/qZ}(K)) in Ψ^{+}.

The automorphism u ∈ Ψ^{+} acts isometrically on the Blanchfield pairing b : M^{+} × M^{+} → ℚ(t)/Λ^{+} (i.e., b(ug, uh) = b(g, h) for all g, h ∈ M).

Let Z = D^{4}−N(K) be the complement of an open regular neighbourhood of an equivariant null concordance K for K. Then H_{∗}(Z; Λ^{+}) and H_{∗}(Z, ∂Z; Λ^{+}) are finitely generated Λ^{+}-torsion modules. The inclusion of X into ∂Z = X ∪(S^{1} ×D^{2}) induces an isomorphism M ≅ H_{1}(∂Z; Λ^{+}), and the Blanchfield pairings agree. Let P be the image of H_{2}(Z, ∂Z; Λ^{+}) in H_{1}(∂Z; Λ^{+}). Then P is invariant under the automorphism u, and so is a Ψ^{+}-submodule.

By Lemmas 2 and 3 we have P^{+} = Ψ^{+} ⊗_{ℤ[ \frac{1}{q} ,z]} P̃, where P̃ is finitely generated and projective as a Γ^{+}-module. Let P̂ be a projective complement: P̃ ⊕ P̂ ≅ (Γ^{+})^{g} for some g ≥ 0. Let P_{1} = ℤ[\frac{1}{q}, z] ⊗_{ℤ} P̃ and P_{2} = ℤ[\frac{1}{q}, z] ⊗_{ℤ} P̂, with the ℤ[\frac{1}{q}, z] ⊗_{ℤ} Γ^{+}-actions given by z^{m}u^{n}(r ⊗ p) = z^{m}r ⊗ u^{n}p, for all r ∈ ℤ[\frac{1}{q}, z] and p ∈ P̃ or P̂, respectively. Then P_{1} ⊕ P_{2} ≅ (ℤ[\frac{1}{q}, z] ⊗_{ℤ} Γ^{+})^{g}. Let D(p_{1}, p_{2}) = ((z − 1)p_{1}, p_{2}) for all (p_{1}, p_{2}) ∈ P_{1} ⊕ P_{2}. Then

0 → (ℤ[\frac{1}{q}, z] ⊗_{ℤ} Γ^{+})^{g} ⟶^{D} (ℤ[\frac{1}{q}, z] ⊗_{ℤ} Γ^{+})^{g} → P̃ → 0

is exact. Extending coefficients to Ψ^{+} gives a square presentation matrix for P^{+}, and so E_{0}(P^{+}) = (α), where α = det(D).

Let χ : Ψ^{+} → Λ^{+} be the Λ^{+}-linear function defined by χ(Σr_{j}u^{j}) = r_{0}. If N is a Ψ^{+}-module let N|_{Λ^{+}} denote the underlying Λ^{+}-module. Composition with χ induces natural isomorphisms Hom_{Ψ^{+}}(N, Ψ^{+}) ≅ Hom_{Λ^{+}}(N|_{Λ^{+}}, Λ^{+}) for all finitely generated Ψ^{+}-modules. These are Ψ^{+}-linear if we set uϕ(n) = ϕ(un) for all ϕ : N → Λ^{+} and n ∈ N. Since restriction is exact and takes projectives to projectives these lead to natural isomorphisms Ext_{Ψ^{+}}^{i}(N, Ψ^{+}) ≅ Ext_{Λ^{+}}^{i}(N|_{Λ^{+}}, Λ^{+}) for all i ≥ 0. In particular, if e(P^{+}) = Ext_{Ψ^{+}}^{1}(P^{+}, Ψ^{+}) then e(P^{+}) ≅ Hom_{Λ^{+}}(P^{+}|_{Λ^{+}}, ℚ(t)/Λ^{+}). Moreover this is an isomorphism of Ψ^{+}-modules if we set uϕ(p) = ϕ(up) for all p ∈ P^{+} and ϕ : P^{+} → ℚ(t)/Λ^{+}. (This is the functorial Γ-action via P). Since the transpose D^{tr} is a (square) presentation matrix for e(P^{+}) we have E_{0}(e(P^{+})) = (det(D^{tr})) = E_{0}(P^{+}).

The isomorphism M/P ≅ \overline{e(P)} implicit in Theorem 2.3 of [Hi] is Ψ-linear. The module M^{+} has a presentation matrix which is block-triangular, with diagonal blocks D and \overline{D^{tr}}. Hence (Δ_{Z/qZ}(K)) = (Δ_{0}(M)) = (Δ_{0}(M^{+})) = (det(D)\overline{det(D)}) = (α\overline{α}), and so Δ_{Z/qZ}(K) = vt^{n}α\overline{α} for some v ∈ (Γ^{+})^{×} and n ∈ ℤ (since (Ψ^{+})^{×} = (Γ^{+})^{×}⟨t⟩). Since Δ_{Z/qZ}(K) has been normalized 1 = Δ_{1}(\overline{A}) = vα(1, u)α(1, u^{−1}) and n = 0, by Corollary 5.3.1 of [Hi]. Hence α(1, u) ∈ Γ^{×}, and so Δ_{Z/qZ}(K) = β\overline{β}, where β = β(t, u) = α(t, u)/α(1, u). We may clearly write β = a(t, u)/q^{m} for some a = a(t, u) ∈ ℤ[t, t^{−1}, u]/(u^{q}−1) such that a(1, u) = q^{m}. □

In particular, Δ_{1}(K)(t) = Π_{ζ^{q}=1}a(t, ζ)a(t^{−1}, ζ^{−1}). Does P have a square presentation matrix over Ψ (i.e., before localization)? Does this theorem together with the original Davis-Naik result imply that Δ_{Z/qZ}(K) = a\overline{a} for some a = a(t, u) ∈ Ψ such that a(1, u) = 1?

The theorem extends easily to equivariantly slice knots with period p in phomology 3-spheres. This special case may be useful in an inductive argument; it is also of interest algebraically, as we may then view Ψ as a pullback of Λ and Λ[ζ_{p}] over 𝔽_{p}Λ, by Rim’s Theorem (cf [DL90]).

Acknowledgment. I would like to acknowledge the support of Grey College and the Department of Mathematical Sciences at the University of Durham (through a Grey College Mathematics Fellowship) while writing this paper.

References

[DL90] Davis, J.F. and Livingston, C. Alexander polynomials of periodic knots, Topology 30 (1991), 551-564.

[DN02] Davis, J.F. and Naik, S. Alexander polynomials of equivariant slice and

ribbon knots, arXiv:math.GT/0201179 (19 January 2002).

[Hi] Hillman, J.A. Algebraic Invariants of Links, World Scientific Publishing Co., Singapore (2002).