SPINNING AND BRANCHED CYCLIC COVERS OF KNOTS. 1. Introduction

Transcription

1 SPINNING AND BRANCHED CYCLIC COVERS OF KNOTS C. KEARTON AND S.M.J. WILSON Abstract. A necessary and sufficient algebraic condition is given for a Z- torsion-free simple q-knot, q >, to be the r-fold branched cyclic cover of a knot. This result is then used to give a necessary and sufficient algebraic condition on a simple (q )-knot for its spun knot to be the r-fold branched cyclic cover of a knot. Finally we give examples for q odd of a knot which is not the r-fold branched cyclic cover of any knot although its spun knot is.. Introduction An n-knot k is a locally-flat oriented PL pair ( S n+, S n). Let K denote the closed complement of a regular neighbourhood of S n in S n+. A simple n-knot k is an n-knot such that K has the homotopy type of a circle up to the middle dimension, i.e. has the [(n )/]-type of a circle. The simple (q )-knots, q >, have been classified in terms of the knot module and Blanchfield pairing in [8] and by H.F. Trotter in [6, 7]. They were originally classified in terms of the Seifert matrix by J. Levine in []. The simple q-knots, q 3, have been classified in terms of the F-form in [9, 7] and, more generally, in [, ]. The following facts are well-known in knot theory. S n has a regular neighbourhood PL homeomorphic to S n B ; let K denote the closed complement of this neighbourhood. Then by the result of Hurewicz, π (K) abelianises to H (K), which by Alexander duality is isomorphic to (t :), the infinite cyclic group written multiplicatively with a preferred generator t determined by the orientations and the duality isomorphisms. Let K be the infinite cyclic cover of K corresponding to the commutator subgroup of π (K); then (t :) acts on K as the group of covering transformations, and H ( K) is a Z [ t, t ] -module. Let K r denote the cover of K corresponding to the kernel of the map π (K) H (K) = (t :) (t : t r ); then K r is the r-fold cyclic cover of K. Since K r = S n S, being an r-fold cover of S n S, we may set K r = K ( r S n B ) to obtain the r-fold cover of S n+ branched over S n. It may happen that K r = S n+, in which case we have another n-knot k r, which we refer to as the r-fold branched cyclic cover of k. In this case H ( K) is a Z [t r, t r ]-module when K is regarded as the infinite cyclic cover of K r. Note that k r is the fixed point set of the Z r action on S n+ = K r given by the covering transformations. We give a necessary and sufficient condition on the F-form of a Z-torsion-free simple q-knot, q 3, for it to be the r-fold branched cyclic cover of some knot (Theorem.5). This is based on the corresponding result of P.M. Strickland [5] for the odd-dimensional case. There is a construction due to E. Artin, known as spinning, which when applied to a simple (q )-knot k yields a simple q-knot σ(k). The result above is 99 Mathematics Subject Classification. 57Q45, E39. Key words and phrases. Simple knot, spun knot, branched cyclic cover, hermitian form, isometry.

2 C. KEARTON AND S.M.J. WILSON used to give a necessary and sufficient condition on the knot module of a simple (q )-knot k for its spun knot σ(k) to be the r-fold branched cyclic cover of a knot (Theorem 3.4). Finally we give examples of simple (4q + )-knots k which are not the r-fold branched cyclic covers of any knot but for which σ(k) is.. Branched cyclic covers First in this section we state the first classification theorem referred to in the introduction; see [8] for details. Let Λ t = Z [ t, t ], so that Q(t) is the field of fractions of Λ t. Define conjugation in Λ t to be the linear extension of t = t. Theorem.. Let k be a simple (q )-knot and let A t = H q K. Then () A t is a finitely generated Λ t -torsion module; () ( t): A t A t is an automorphism; (3) there is a non-singular ( ) q+ -hermitian pairing <, > t : A t A t Q(t)/Λ t where non-singular means that the adjoint map A t Hom (A t, Q(t)/Λ t ) is an isomorphism. If q = then the signature is divisible by 6. For q >, k is determined by the pair (A t, <, > t ), and for q any module and pairing satisfying the conditions above can be realised by a simple (q )-knot, provided that if q = the signature is divisible by 6. Now let us recall the work of Paul Strickland [5]. Suppose that the simple (q )- knot k = ( S q+, S q ), q, is the r-fold branched cyclic cover of some knot l, that is k = l r. Then l is necessarily a simple knot since the universal cover of its exterior is K, but the group of covering transformations is (u :) where u r = t. Thus A t can be regarded as a Λ u -module, A u say, with Blanchfield pairing <, > u. The Blanchfield pairing is defined as follows. First choose a triangulation of the exterior of l; this lifts to a triangulation of K which is invariant under the action of u (and hence of t). Let C q denote the group of q-chains, C q+ the group of (q + )- chains in the dual triangulation. Take a, b A t ; as A t is a Λ t -torsion module, there exists a non-zero π(t) Λ t such that πa = 0. Thus there exists α C q+ whose boundary represents πa. Let β C q represent b. Then the Blanchfield pairing of k is defined by () a, b t = and that of l is given by () a, b u = ( i= ( i= I ( α, t i β ) t i ) I ( α, u i β ) u i ) π(t) Q(t) Λ t π(u r ) Q(u) Λ u where I ( α, u i β ) Z is the algebraic intersection of the chain and dual chain. By grouping the integers into congruence classes (mod r), Strickland [5] shows that a, b u = m M u k µ ( a, u m b t )

3 SPINNING AND BRANCHED CYCLIC COVERS OF KNOTS 3 where M is a complete set of residues (mod r) and Q(t) µ: Q(u) Λ t Λ u is f(t) f (u r ). He proves the following result. Theorem.. Let k be a simple (q )-knot, q. Then k is the r-fold branched cyclic cover of a knot if and only if there exists an isometry u of (A t, <, > t ) such that u r = t. The rest of this section is concerned with extending this result to the even-dimensional Z-torsion-free case. First we recall the classification theorem of [9]. Let k be a simple q-knot, q 4, such that H q = H q K is Z-torsion-free. Then there is a hermitian pairing, that is, linear in the first variable and conjugate linear in the second, (3) <, >: H q+ H t Q(t)/Λ t where H q+ = H q+ K and the pairing is non-singular in the sense used above. This is the standard Blanchfield pairing, defined in a similar way to () above. It turns out that Q(t)/Λ t is too large for our purposes, so we note that <, > takes its values in Λ S /Λ t where S = {f(t) Λ t : f() = ±} and Λ S = { f g : f Λ t, g S [ Set Γ t = Z ] t, t ; thus Γ t may be regarded as Λ t with coefficients reduced mod. Let Γ t denote the field of fractions of Γ t. Let H i = H i /H i for i = q, q +, and let Π q+ = π q+ K /π q+ K. It is shown in [9] that there is a canonical short exact sequence of Γ t modules and a non-singular hermitian pairing }. E : 0 H q Ω Πq+ H Hq+ 0 [, ] t : Π q+ Π q+ Γ t Γ t satisfying Ha, b t = [a, Ωb] t where a Π q+, b H q, and <, > t is obtained from (3) above by reducing coefficients mod. The pairing [, ] t is defined in the same way as the Blanchfield pairing in (), but using a homotopy intersection T in place of I (see [9, pages 5 0] for details). All this is summed up as follows. Following Levine [3] we say that a Λ t -module is of type K is it is finitely generated and multiplication by t induces an automorphism. Definition.3. An F-form (E, H q, p q, [, ] t, <, > t ) consists of the following. () A short exact sequence of Γ t modules () A non-singular hermitian pairing E : 0 H q Ω Πq+ H Hq+ 0 [, ] t : Π q+ Π q+ Γ t Γ t

4 4 C. KEARTON AND S.M.J. WILSON (3) A non-singular hermitian pairing <, > t : H q+ H q Γ t Γ t related to [, ] t by Ha, b t = [a, Ωb] t for all a Π q+, b H q. (4) A Z-torsion-free Λ t -module H q of type K. (5) A short exact sequence of Λ t -modules p q 0 H q H q Hq 0 where H q is regarded as a Λ t -module via the ring homomorphism Λ t Γ t given by reducing coefficients mod. An isometry of F-forms is defined in the obvious way. In [9] the following is proved. Theorem.4. A Z-torsion-free simple q-knot, q 4, gives rise to an F-form. Two such knots with isometric F-forms are ambient isotopic. Finally, every F-form arises from some Z-torsion-free simple q-knot, for each q 4. In [7] this result is extended to the case q = 3. A careful reading of [5] shows that the same proofs apply, almost verbatim, to the case of Z-torsion-free simple q-knots, and we can state the following result. Theorem.5. Let k be a Z-torsion-free simple q-knot, q 3. Then k is the r-fold branched cyclic cover of a knot if and only if there exists an isometry u of the F-form of k such that u r = t. Moreover, the pairings are related by a, b u = m M u m µ ( a, u m b t ) [a, b] u = m M u m µ ([a, u m b] t ) where M is a complete set of residues (mod r) and µ (f(t)) = f (u r ) in Λ t or Γ t. 3. Simple spun knots The papers [4, 5, 6] of C.McA. Gordon form an excellent reference for spinning knots, and we shall draw on his descriptions and results without further comment. First we recall a definition of spinning. Let k be the n-knot ( S n+, S n) and let B be a regular neighbourhood of a point on S n such that (B, B S n ) is an unknotted ball pair. Then the closure of the complement of B in S n+ is a knotted ball pair ( B n+, B n), and σ(k) is the pair [( B n+, B n) B ]. Lemma 3.. Let k be an n-knot and r an integer such that the r-fold cyclic cover of S n+ branched over k is a sphere. Then the r-fold cyclic cover of S n+3 branched over σ(k) is also a sphere, and σ (k r ) = σ(k) r. Proof. Using the notation above, K r is the union along the boundary of the r-fold cyclic cover of B branched over B S n, which is just an n+-ball since (B, B S n ) is unknotted, and the r-fold cyclic cover of B n+ branched over B n. The latter must therefore also be an n + -ball since K r is a sphere. Let us call this knotted ball pair ( B n+, B n). Then σ (k r r) = [( B n+, B n) B], which is the boundary of r the r-fold cyclic cover of B n+ B branched over B n B, that is, σ(k) r.

5 SPINNING AND BRANCHED CYCLIC COVERS OF KNOTS 5 Next we recall the results of [0]. Definition 3.. An F-form (E, H q, p q, [, ] t, <, > t ) is hyperbolic if the short exact sequence E is split, so that there is a short exact sequence i 0 H j q+ Πq+ Hq 0 such that j Ω =identity, Hi =identity; Im i is self-annihilating under [, ] t. Thus Π q+ = Im Ω Im i, the direct sum of two self-annihilating submodules. The following is proved in [0]. Theorem 3.3. Let l be a simple (q )-knot, q 5. Then the F-form of σ(l) is hyperbolic. Now we can state the main result of this section. Theorem 3.4. Let k be a simple (q )-knot, q 5. Then σ(k) is the r-fold branched ( cyclic ) cover of a knot if and only if there is a Λ t -module isomorphism u: H q K H q K such that u r = t. Proof. Let (E, H q, p q, [, ] t, <, > t ) be the F-form of σ(k), so that H q = Hq K. The existence of such an isomorphism is clearly necessary for σ(k) to be the r-fold branched cyclic cover of a knot, by Theorem.5. To prove the converse, assume that we are given an automorphism u of H q such that u r = t; then u induces an automorphism, also denoted u, of H q. Using the fact that <, > t is non-singular, we define u: H q+ H q+ by < ua, ub > t =< a, b > t, a H q+, b H q Clearly this is an automorphism. Using the fact that the F-form is hyperbolic, by Theorem 3.3, we can now define u on Π q+ by uωb = Ωub if b H q ui a = i ua if a H q+ Then for all a, c H q+, b, d H q we have [u(i a + Ωb), u(i c + Ωd)] t = [i ua + Ωub, i uc + Ωud] t = [i ua, Ωud] t + [i uc, Ωub] t =< Hi ua, ud > t +< Hi uc, ub > t =< ua, ud > t +< uc, ub > t =< a, d > t +< c, b > t =< Hi a, d > t +< Hi c, b > t = [i a, Ωd] t + [i c, Ωb] t = [i a + Ωb, i c + Ωd] t and so u is an isometry of the F-form of σ(k). By Theorem.5, σ(k) = l r for some knot l.

6 6 C. KEARTON AND S.M.J. WILSON 4. Examples Now we give some examples of a simple (4q + )-knot k such that σ(k) is the -fold branched cyclic cover of a knot but k is not. Example 4.. Let τ = m+ɛ+ D m where m Z, ɛ = ±, and D = 4mɛ + is not a square. Then τ is a root of the polynomial f(t) = mt (m + ɛ)t + m. Assume that there is a prime p 3 mod 4 dividing D (e.g. m =, ɛ = [, p = 3, or m =, D ] ɛ =, p = 7). Let be the non-trivial automorphism of Q ; we shall refer [ D ] to this as conjugation in Q. Note the following. () τ τ =, so [ τ = τ. ] () Let R = Z m, + D = Z [ τ, τ ]. ( (3) Let P be the ideal p, ) D. We have a ring homomorphism ψ from R to R F p, the field with p elements, which has kernel P. We will refer to this as reduction mod P and all our congruences will be mod P. (4) 4mɛ (mod p and hence mod P ). So τ = 4mɛ + + ɛ D ( + ) 4mɛ and this is not a square mod P (i.e. mod p). (5) For a R, a ã. Now let M = R as a Λ t -module, the action of t being multiplication by τ, and define a hermitian form on R by (x, y) = xỹ. By property (), this corresponds to a non-singular hermitian form on M as a Λ t -module, given by g(t)h ( t ) g(t), h(t) t = mt (m + ɛ) + mt (g(τ), h(τ)) = g(τ)h ( τ). Multiplication by (t ) gives an automorphism of M because f() = ±, and because by the Remainder Theorem (t ) divides f(t) f(). Alternatively we can note that ɛ = (τ )(mτ m ɛ). By Theorem. there is a unique simple (4q + )-knot with this as its module and pairing, for each q. Let M be a copy of M, but with pairing given by (x, y) = xỹ, and let N be the orthogonal direct sum of M and M. Thus N = R R with the action of t being given by τ 0 T = 0 τ and the hermitian form by ( ) 0 It is easy to see that T is the square of the R-module isomorphism 0 τ 0 of N but we shall now show that T is not the square of an isomorphism of N which is also an isometry of the hermitian form 0. For suppose that A were such an isometry. Then a b A = GL c d (R)

7 SPINNING AND BRANCHED CYCLIC COVERS OF KNOTS 7 and for A to be an isometry (4) A 0 Ã = This multiplies out to aã c c a b c d ãb cd b b d d = 0 0 So equation (4) gives, working mod P (and using property (5) above), By (i) and (ii), (i) a c, (ii) b d, (iii) ab cd. (iv) a and d 0 since is not a square mod p. Choose v Z so that Then from (iii), In particular so that, since T is to be A, τ 0 0 τ Thus (since τ ) (v) d va. (vi) b vc. a vc A c va a + vc avc + v ca ac + vac vc + v a. (vii) a + vc, (viii) vca( + v) 0. Now since, by (iv), a 0, (viii) shows that vc 0 or v. But vc 0 implies, by (vii), that a, which is not so. Thus v. But now (i) and (vii) give, which is not so. Thus no such isometry exists. Example 4.. Now we specialise to the case m =, ɛ =, p = 3, so that f(t) = t+t, the Alexander polynomial of the trefoil knot. Note that f(t) is the 6 th cyclotomic polynomial. For any integer q > 0, there exists a simple (4q + )- knot k as in Example 4. with knot module N. By the results of Gordon in [3], the r-fold branched cyclic cover of k is a sphere for every r mod 6. And by Theorem., k r is the same as k. Thus for any r N, r mod 6, and for any integer q > 0, there exists a simple (4q + )-knot k such that σ(k) is the r-fold branched cyclic cover of some knot although k is not. Here is one last example, much simpler than the ones above but not capable of generalising to the r-fold case. Example 4.3. Let f(t) = t 3t +, which has roots 3± 5. Set τ = 3+ 5, ξ = + 5, and note that ξ = τ. Put R = Z [ξ] = Z [ τ, τ ] and define conjugation in the obvious way by ξ = 5. Think of R as an R-module, and put a hermitian form on it by setting (x, y) = xỹ. Since ξ ξ =, ξ is an isomorphism on R but not an isometry. Since τ only has two square roots, ±ξ, there are no isometries whose square is τ. In the usual way, (R, (, )) corresponds to a knot module and pairing for a simple (4q + )-knot, q > 0.

8 8 C. KEARTON AND S.M.J. WILSON References [] M.S. Farber, An algebraic classification of some even-dimensional knots. I, Trans. Amer. Math. Soc. 8 (984), [] M.S. Farber, An algebraic classification of some even-dimensional knots. II, Trans. Amer. Math. Soc. 8 (984), [3] C.McA. Gordon, Knots whose branched cyclic coverings have periodic homology, Trans. Amer. Math. Soc. 68 (97), [4] C.McA. Gordon, Some higher-dimensional knots with the same homotopy groups, Quart. Jour. Math. Oxford 4 (973), 4 4. [5] C.McA. Gordon, On the higher-dimensional Smith conjecture, Proc. Lond. Math. Soc. 9 (974), [6] C.McA. Gordon, A note on spun knots, Proc. Amer. Math. Soc. 58 (976), [7] J.A. Hillman and C. Kearton, Seifert matrices and 6-knots, Trans. Amer. Math. Soc. 309 (988), [8] C. Kearton, Classification of simple knots by Blanchfield duality, Bull. Amer. Math. Soc. 79 (973), [9] C. Kearton, An algebraic classification of certain simple even-dimensional knots, Trans. Amer. Math. Soc. 76 (983), 53. [0] C. Kearton, Simple spun knots, Topology 3 (984), [] C. Kearton and S.M.J. Wilson, Cyclic group actions on odd-dimensional spheres, Comm. Math. Helv. 56 (98), [] J. Levine, An algebraic classification of some knots of codimension two, Comm. Math. Helv. 45 (970), [3] J. Levine, Knot modules I, Trans. Amer. Math. Soc. 9 (977), 50. [4] H. Seifert, Über das Geschlect von Knoten, Math. Ann. 0 (934), [5] Paul Strickland, Branched cyclic covers of simple knots, Proc. Amer. Math. Soc. 90 (984), [6] H.F. Trotter, On S-equivalence of Seifert matrices, Invent. math. 0 (973), [7] H.F. Trotter, Knot modules and Seifert matrices, Knot Theory, Ed. J.C. Hausmann, LNM 685, Springer-Verlag, New York-Heidelberg-Berlin, 978. Mathematics Department, University of Durham, South Road, Durham DH 3LE, England. address: Cherry.Kearton@durham.ac.uk, S.M.J.Wilson@durham.ac.uk