Knots having the same Seifert form and primary decomposition of knot concordance
KKnots having the same Seifert form and primarydecomposition of knot concordance
Taehee Kim
Abstract.
We show that for each Seifert form of an algebraically slice knot with nontrivialAlexander polynomial, there exists an infinite family of knots having the Seifert form such thatthe knots are linearly independent in the knot concordance group and not concordant to anyknot with coprime Alexander polynomial. Key ingredients for the proof are Cheeger–Gromov–vonNeumann ρ (2) -invariants for amenable groups developed by Cha–Orr and polynomial splittingsof metabelian ρ (2) -invariants.
1. Introduction
A knot is slice if it bounds a locally flat 2-disk in the 4-ball, and two knots K and J are concordant if K − J ) is slice. The concordance classes of knots form an abeliangroup under connected sum. This abelian group is called the knot concordance group ,which we denote by C . There is a surjective homomorphism from C to the algebraicconcordance group of Seifert forms which sends the concordance class of a knot to thealgebraic concordance class of a Seifert form of the knot. It is known by Levine [Lev69a,Lev69b] and Stoltzfus [Sto77] that the algebraic concordance group is isomorphic to Z ∞ ⊕ Z ∞ ⊕ Z ∞ . The kernel of the above surjection is the subgroup of (the concordance classesof) algebraically slice knots , denoted A . The classification of the group A (and C ) is yetunknown, and in this paper we address the structure of A related to Seifert forms andthe Alexander polynomial. Note that a knot with trivial Alexander polynomial is slice bythe work of Freedman [Fre82, FQ90]. Theorem 1.1 (Main Theorem) . Let V be a Seifert form of an algebraically slice knot K with nontrivial Alexander polynomial ∆ K ( t ) . Then there exists an infinite family of knots { K i } i ∈ N which satisfies the following: (1) the knots K i have the Seifert form V , (2) the knots K i are linearly independent in C , (3) for each i and nonzero integer n , the knot nK i is not concordant to any knotwhose Alexander polynomial is coprime to ∆ K ( t ) . We review known results on the structure of knot and link concordance under fixedAlexander invariants and primary decomposition of C . The aforementioned work of Freed-man is equivalent to that if ∆ K ( t ) = 1, then K is concordant to the unknot. Namely,trivial Alexander polynomial determines a unique concordance class. A natural questionarises asking if there is any other Alexander polynomial or a Seifert form which determinesa unique concordance class. This question was answered in the negative by Livingston[Liv02] using Casson–Gordon invariants under a certain condition on Seifert forms. Later,using Cheeger–Gromov–von Neumann ρ (2) -invariants, the author removed the conditionon Seifert forms and gave the following theorem: Theorem 1.2. [Kim05b]
Let V be a Seifert form of a knot K with nontrivial Alexanderpolynomial. Then, there exists an infinite family of knots { K i } i ∈ N such that K i have theSeifert form V and are pairwise nonconcordant. Mathematics Subject Classification.
Key words and phrases.
Knot, Concordance, Seifert form, Amenable signature. a r X i v : . [ m a t h . G T ] A ug NOTS HAVING THE SAME SEIFERT FORM 2
In [Kim05b] it was not shown that the K i in Theorem 1.2 are linearly independentin C , and Theorems 1.1(1) and (2) extend Theorem 1.2 for the case of Seifert forms ofalgebraically slice knots by giving examples which are linearly independent in C .Theorem 1.2 was extended in various directions. Cochran and the author [CK08]gave an infinite family of pairwise nonconcordant knots having the same higher-order Alexander invariants, and it was extended further in [Kim16] so that the knots are linearlyindependent in C . Cha, Friedl, and Powell [CFP14] generalized Theorem 1.2 to linkconcordance. Recently Kauffman and Lopes gave infinitely many nonisotopic pretzelknots with the same Alexander invariants [KL16].Theorem 1.1(3) is related to primary decomposition of knot concordance. A theoremof Levine [Lev69a] playing an essential role in classification of the algebraic concordancegroup is that if the connected sum of two knots with coprime Alexander polynomials isalgebraically slice, then so are the knots. A similar decomposition in C or A is unknown,and we have the following open question: if two knots K and J have coprime Alexanderpolynomials and the connected sum K J is slice, then are K and J slice? Put differently,if K and J have coprime Alexander polynomials and any of K and J is not slice, then is K nonconcordant to J ?Regarding the above question, Se-Goo Kim [Kim05a] showed splittings of Casson–Gordon invariants for knots with coprime Alexander polynomials. A similar polynomialsplitting property of the metabelian Cheeger–Gromov–von Neumann ρ (2) -invariants wasshown by Se-Goo Kim and the author [KK08] (see Theorem 3.2), and it was extendedto splittings of higher-order ρ (2) -invariants [KK14]. On smooth concordance, similarpolynomial splittings of d -invariants on slicing knots [Bao15] and doubly slicing knots[KK16] were also shown.In [Kim05a, KK08], the examples of knots which are not concordant to any knot withcoprime Alexander polynomial were given, but they were constructed for some prescribedAlexander modules. For instance, the examples in [KK08] have Alexander modules whichhave a unique nontrivial proper submodule. On the other hand, Theorem 1.1 gives ex-amples for any Seifert form of an algebraically slice knot with nontrivial Alexander poly-nomial.There is the solvable filtration {F n } of C defined in [COT03], which is indexed bynonnegative half-integers. A notable property of {F n } is that all metabelian slicenessobstructions, including Casson–Gordon invariants, vanish for knots in F n when n ≥ . n , Cochran, Harvey, and Leidy [CHL11b] gave a similar primarydecomposition of a family of knots in F n constructed using robust doubling operators (seeDefinitions 4.4 and 7.2 and Theorem 7.7 in [CHL11b]). Also, there is a similar primarydecomposition of a family of order 2 elements in F n [CHL11a, Jan15]. In [CHL11b,CHL11a, Jan15], the examples of knots were shown to be nonconcordant to any knotwith coprime Alexander polynomial which is constructed using doubling operators andArf invariant zero knots (for instance, see [CHL11b, Theorem 6.2]).To construct K i in Theorem 1.1, we use (iterated) satellite construction. To showtheir linear independence in C , we use Cheeger–Gromov–von Neumann ρ (2) -invariants foramenable groups, which were developed by Cha and Orr [CO12] on homology cobordism,and later adapted to knot concordance by Cha [Cha14] (see Theorem 2.1). To showTheorem 1.1(3), we use polynomial splittings of metabelian ρ (2) -invariants in [KK08] (seeTheorem 3.2).This paper is organized as follows. We review necessary results on ρ (2) -invariants inSection 2. In Section 3, we give a proof of Theorem 1.1. In this paper, homology groupsare with integer coefficients unless specified otherwise. By abuse of notation, we use the NOTS HAVING THE SAME SEIFERT FORM 3 same symbol for a knot and its homology and homotopy classes. For a prime p , we denotethe field of p elements by Z p . All manifolds are assumed to be oriented and compact. Acknowledgments
This research was supported by Basic Science Research Program through the NationalResearch Foundation of Korea(NRF) funded by the Ministry of Education (no. 2011-0030044(SRC-GAIA) and no. 2015R1D1A1A01056634).
2. Preliminaries
In this section, we review necessary results on ρ (2) -invariants in [COT03, Cha14, KK08].Let M be a closed 3-manifold and φ : π M → Γ a homomorphism to a countable(discrete) group Γ. By enlarging the group Γ if necessary, we may assume that thereexists a 4-manifold W with ∂W = M such that φ extends to ˜ φ : π W → Γ. Then, the Cheeger–Gromov–von-Neumann ρ (2) -invariant associated with ( M, φ ) [CG85] can bedefined to be the L -signature defect as follows: ρ (2) ( M, φ ) := sign (2)Γ ( W ) − sign( W ) . In the above, sign( W ) is the ordinary signature of W , and sign (2)Γ ( W ) is the L -signature ofthe intersection form on H ( W ; N Γ) where N Γ denotes the group von Neumann algebraof Γ. We refer the reader to [Cha14, Section 2] for more details on ρ (2) -invariants. Basedon the work on ρ (2) -invariants in [CO12], Cha obtained the following sliceness obstruction,which extends the sliceness obstruction using ρ (2) -invariants in [COT03]. In the following, M ( K ) denotes the zero-framed surgery on a knot K in S . Theorem 2.1. [Cha14, Theorem 1.2]
Suppose K is a slice knot and Γ is an amenablegroup lying in Strebel’s class D ( R ) for some ring R . If φ : π M ( K ) → Γ is a homomor-phism extending to a slice disk exterior for K , then ρ (2) ( M ( K ) , φ ) = 0 . One can find the definitions of amenable group and Strebel’s class D ( R ) in [CO12],but they will not be needed in this paper; we will only need Lemma 2.2 below.For a group G , let G (1) := [ G, G ], the commutator subgroup of G , and let G (2) :=[ G (1) , G (1) ]. For a prime p , we also define G (2) p := Ker { G (1) −→ ( G (1) /G (2) ) ⊗ Z p } . Lemma 2.2.
Suppose G is a group with H ( G ) ∼ = Z . Then, for each prime p , the group G/G (2) p is amenable and lies in Strebel’s class D ( Z p ) .Proof. Since
G/G (1) ∼ = H ( G ) ∼ = Z and G (1) /G (2) p injects into ( G (1) /G (2) ) ⊗ Z p , the groups G (1) /G (2) and G (1) /G (2) p are abelian and have no torsion coprime to p . Now the conclusionfollows from [CO12, Lemma 6.8]. (cid:3) We review a vanishing criterion for metabelian ρ (2) -invariants for slice knots in[COT03].For a knot K , there is the rational Blanchfield form B(cid:96) : H ( M ( K ); Q [ t ± ]) × H ( M ( K ); Q [ t ± ]) −→ Q ( t ) / Q [ t ± ] . For a Q [ t ± ]-module P of H ( M ( K ); Q [ t ± ]), we define P ⊥ := { x ∈ H ( M ( K ); Q [ t ± ]) | B(cid:96) ( x, y ) = 0 for all y ∈ P } . NOTS HAVING THE SAME SEIFERT FORM 4 If P = P ⊥ , we say that P is self-annihilating with respect to the rational Blanchfieldform.Letting the group Z = H ( M ( K )) = (cid:104) t (cid:105) act on H ( M ( K ); Q [ t ± ]) via the actionof t , we obtain the semi-direct product H ( M ( K ); Q [ t ± ]) (cid:111) Z . Then, each element x ∈ H ( M ( K ); Q [ t ± ]) induces a homomorphism φ x : π M ( K ) −→ H ( M ( K ); Q [ t ± ]) (cid:111) Z −→ Q ( t ) / Q [ t ± ] (cid:111) Z such that φ x ( y ) = ( B(cid:96) ( x, yµ − (cid:15) ( y ) ) , (cid:15) ( y )) where µ is the meridian of K and (cid:15) : π M ( K ) → Z = H ( M ( K )) is the abelianization (see [COT03, Section 3]).We say that K has vanishing metabelian ρ (2) -variants if there exists a self-annihilatingsubmodule P with respect to the rational Blanchfield form such that ρ (2) ( M ( K ) , φ x ) = 0for all x ∈ P . We have the following theorem. Theorem 2.3. [COT03, Theorem 4.6]
A slice knot has vanishing metabelian ρ (2) -invariants.
3. Proof of Theorem 1.1
Construction of the K i . Let K be an algebraically slice knot with ∆ K ( t ) (cid:54) = 1 whichhas a Seifert form V . Since there exists a slice knot having the same Seifert form as K (for instance, see [Kaw96, Proposition 12.2.1]), we may assume that K is slice.We will construct the desired K i using (iterated) satellite construction. We brieflyexplain satellite construction we will use in this paper. Let η , η , . . . , η m be simple closedcurves in S (cid:114) K such that the curves η (cid:96) form an unlink in S . Let J be a knot. Nowtake the union of S (cid:114) N ( η ) and S (cid:114) N ( J ) along their common boundary S × S viaan orientation reversing homeomorphism such that a meridian (resp. 0-framed longitude)of η is identified with a zero-framed longitude (resp. a meridian) of J . Iterating thisprocess, for each (cid:96) = 1 , , . . . , m , replace the open tubular neighborhood N ( η (cid:96) ) of η (cid:96) withthe exterior of J . The resulting ambient space is homeomorphic to S , and the image of K under this process becomes a new knot in S , which we denote by K ( η , . . . , η m ; J ) or K ( η (cid:96) ; J ) for simplicity. We will construct K i as K ( η (cid:96) ; J i ) form some choice of η (cid:96) and J i ,where the choice of η , . . . , η m will be independent of i .We choose η , . . . , η m for K i as follows. Let F be a Seifert surface for K with whichthe Seifert form V is associated. Considering F as a disk with 2 g bands added, take η (cid:96) to be the curves dual to the bands of F (hence m = 2 g ).Since H ( M ( K ); Z p [ t ± ]) ∼ = H ( M ( K ); Z [ t ± ]) ⊗ Z p and Z p is a field, it easily followsthat the η (cid:96) generate H ( M ( K ); Z p [ t ± ]) for each prime p . It is well-known that the η (cid:96) also generate H ( M ( K ); Q [ t ± ]). This is a key property of the η (cid:96) which we will use later.We explain how to choose J i . In [CG85], it was shown that there exists a constant C K such that | ρ (2) ( M ( K ) , φ ) | < C K for every homomorphism φ : π M → Γ where Γ is acountable group. (One can take an explicit value for C K as 69713280 · c ( K ) where c ( K )is the crossing number of K [Cha16, Theorem 1.9].) Now we choose J i to be the knotsin Lemma 3.1 below. For a knot K , let a K be the top coefficient of ∆ K ( t ) and σ K theLevine-Tristram signature function for K . Lemma 3.1.
For the constants C K and a K defined as above, there exists a sequence ofknots J , J , . . . and a sequence of primes p , p , . . . which satisfy the following: (1) Arf( J i ) = 0 for each i and a K < p < p < p < · · · , (2) (cid:80) p i − r =0 σ J i ( e πr √− /p i ) > p i C K for all i , (3) (cid:80) p i − r =0 σ J j ( e πr √− /p i ) = 0 for j > i , (4) (cid:82) S σ J i ( ω ) dω > C K for all i . NOTS HAVING THE SAME SEIFERT FORM 5
Proof.
Let { p i } be any increasing sequence of primes bigger than a K . Let w i := e π √− /p i .By [Cha09, Lemma 5.6], for each i there exists a knot L i and neighborhoods N ( ω i ) and N ( ω − i ) of ω i and ω − i , respectively, which are disjoint from ω rj for all j < i and all r ∈ Z such that σ L i is positive inside N ( ω i ) ∪ N ( ω − i ) and 0 outside N ( ω i ) ∪ N ( ω − i ). Now foreach i , the desired knot J i can be obtained by taking the connected sum of sufficientlymany even number of copies of L i . (cid:3) Now for each i we define K i := K ( η (cid:96) ; J i ) where η (cid:96) and J i are defined as above. Proof of Theorem 1.1(1).
Since we have chosen η (cid:96) in the complement of the Seifertsurface F for K , for each i , the image of F under the satellite construction for K i becomesa Seifert surface for K i which has the same Seifert form as F . This proves Theorem 1.1(1). Proof of Theorem 1.1(3).
We prove Theorem 1.1(3) before proving Theorem 1.1(2).Recalling Theorem 2.3 and the fact that a knot and its inverse have the same Alexanderpolynomial, we have the following theorem on polynomial splittings of metabelian ρ (2) -invariants. Theorem 3.2. [KK08, Theorem 3.1]
Suppose two knots K and J have coprime Alexanderpolynomials. If K does not have vanishing metabelian ρ (2) -invariants, then K is notconcordant to J . Note that since ∆ nK i ( t ) = (∆ K ( t )) | n | , the Alexander polynomial of a knot is coprimeto that of K if and only if it is coprime to that of nK i . Therefore, by Theorem 3.2, toprove Theorem 1.1(3) it suffices to show that nK i does not have vanishing metabelian ρ (2) -invariants for each n and i . Fix n and i . By taking the inverse of K if necessary,we may assume n >
0. Suppose to the contrary that nK i has vanishing metabelian ρ (2) -invariants. Then, there exists a self-annihilating submodule P of H ( M ( nK i ); Q [ t ± ])such that ρ (2) ( M ( nK i ) , φ x ) = 0 for all x ∈ P . It is well-known that since P is a self-annihilating submodule,rank Q P = 12 rank Q H ( M ( nK i ); Q [ t ± ]) = 12 deg ∆ nK i ( t ) . Since ∆ K ( t ) (cid:54) = 1, it follows that P (cid:54) = 0.Fix x ∈ P such that x (cid:54) = 0. In particular, ρ (2) ( M ( nK i ) , φ x ) = 0 where φ x is thehomomorphism π M ( nK i ) → Q ( t ) / Q [ t ± ] (cid:111) Z induce from x as defined in Section 2. Wewill show that this will lead us to a contradiction.To compute ρ (2) ( M ( nK i ) , φ x ), we construct a cobordism C such that ∂C = M ( nK i ) (cid:97) ( − n (cid:97) M ( K i ))as follows. Let C be the standard cobordism between M ( nK i ) and (cid:96) n M ( K i ) as in[COT04, p.113]. Briefly speaking, C is obtained from (cid:96) n M ( K i ) × [0 ,
1] by attaching n − n copies of K i , and then attaching n − µ j µ − j +1 , respectively, for 1 ≤ j ≤ n − µ j is the meridian ofthe j th copy of K i .Then, one can see that π C ∼ = π ( M ( nK i )) / (cid:104) (cid:96) , . . . , (cid:96) n (cid:105) where (cid:104)· · · (cid:105) denotes the normalsubgroup generated by · · · and each (cid:96) j is the 0-framed longitude of the j th copy of K i (forexample, see [KK14, Lemma 3.1 and p.810]). For simplicity, let G := Q ( t ) / Q [ t ± ] (cid:111) Z .Since (cid:96) j ∈ π ( M ( nK i )) (2) for each j and G (2) = 0, it follows that φ x ( (cid:96) j ) = 0 for all j .Therefore, φ x extends to π C → G , which is also denoted by φ x . NOTS HAVING THE SAME SEIFERT FORM 6
For each j , let φ jx : π ( M ( K i )) → G be the restriction of φ x : π C → G to the j th copy of M ( K i ) in the bottom boundary of C . Let B(cid:96) and
B(cid:96) i denote the ra-tional Blanchfield forms of nK i and K i , respectively. Then, H ( M ( nK i ); Q [ t ± ]) ∼ = ⊕ n H ( M ( K i ); Q [ t ± ]) and B(cid:96) ∼ = ⊕ n B(cid:96) i . Therefore, we can write x = ( x , x , . . . , x n )for some x j ∈ H ( M ( K i ); Q [ t ± ]) for 1 ≤ j ≤ n , and then for each y = ( y , . . . , y n ) ∈ H ( M ( nK i ); Q [ t ± ]) we have B(cid:96) ( x, y ) = (cid:80) nj =1 B(cid:96) i ( x j , y j ). Also, we can identify y j ∈ H ( M ( K i ); Q [ t ± y = ( y , y , . . . , y n ) ∈ H ( M ( nK i ); Q [ t ± ]) such that y i = 0 for i (cid:54) = j , and we obtain B(cid:96) ( x, y j ) = B(cid:96) ( x, y ) = (cid:80) nj =1 B(cid:96) i ( x j , y j ) = B(cid:96) i ( x j , y j ). Therefore,for each j one can deduce that φ jx = φ x j , the homomorphism induced from x j .Now, from the definition of ρ (2) -invariants, we obtainsign (2) G ( C ) − sign( C ) = ρ (2) ( M ( nK i ) , φ x ) − n (cid:88) j =1 ρ (2) ( M ( K i ) , φ x j ) . Using Mayer Vietoris sequences, we can show H ( C ) ∼ = H ( ∂ + C ) where ∂ + C := M ( nK i )),and hence Coker { H ( ∂C + ) −→ H ( C ) } = 0 . Therefore, sign( C ) = 0, and we also obtain sign (2) G ( C ) = 0 by [CO12, Theorem 6.6] (orsee the proof of[COT04, Lemma 4.2]). Therefore, we have ρ (2) ( M ( nK i ) , φ x ) = n (cid:88) j =1 ρ (2) ( M ( K i ) , φ x j ) . Since ρ (2) ( M ( nK i ) , φ x ) = 0 by our choice of x , we obtain(3.1) n (cid:88) j =1 ρ (2) ( M ( K i ) , φ x j ) = 0 . We compute ρ (2) ( M ( K i ) , φ x j ) for each j . If x j = 0, then ρ (2) ( M ( K i ) , φ x j ) = 0. For, inthis case the φ x j maps onto Z , and therefore ρ (2) ( M ( K i ) , φ x j ) = (cid:82) S σ K i ( ω ) dω (see (2.3)on p.108 and Lemma 5.3 in [COT04]). Since K i has the same Seifert form V as the sliceknot K , we have (cid:82) S σ K i ( ω ) dω = 0.Suppose x j (cid:54) = 0. Recall that K i = K ( η (cid:96) ; J i ) = K ( η , . . . , η m ; J i , . . . , J mi ) where J (cid:96)i is the (cid:96) th copy of J i for each (cid:96) = 1 , , . . . , m . Since each longitude (cid:96) j ∈ π ( M ( K i )) (2) ,the homomorphism φ x j uniquely extends to π M ( K ) → G and π M ( J (cid:96)i ) → G for (cid:96) =1 , , . . . , m , which we denote by φ j and φ (cid:96)j , respectively (see [CHL09, p.1429]). Further-more, since the meridian of J (cid:96)i is identified with the longitude of η (cid:96) ∈ π ( M ( K )) (1) , the ho-momorphism φ (cid:96)j maps into G (1) = Q ( t ) / Q [ t ± ], which is an abelian group. Therefore, wehave the following lemma which immediately follows from [CHL09, Lemma 2.3]. For con-venience, let us identify η (cid:96) in M ( K ) with its image in M ( K i ). Note that φ j ( η (cid:96) ) = φ x j ( η (cid:96) ). Lemma 3.3. [CHL09, Lemma 2.3]
In the above setting, we have ρ (2) ( M ( K i ) , φ x j ) = ρ (2) ( M ( K ) , φ j ) + m (cid:88) (cid:96) =1 ρ (2) ( M ( J (cid:96)i ) , φ (cid:96)j ) , where ρ (2) ( M ( J (cid:96)i ) , φ (cid:96)j ) = (cid:40) if φ x j ( η (cid:96) ) = 0 , (cid:82) S σ J i ( ω ) dω if φ x j ( η (cid:96) ) (cid:54) = 0 . NOTS HAVING THE SAME SEIFERT FORM 7
Since the η (cid:96) (1 ≤ (cid:96) ≤ m ) generate H ( M ( K i ); Q [ t ± ]) ∼ = H ( M ( K ); Q [ t ± ]) andthe rational Blanchfield form B(cid:96) i is nonsingular, there exists at least one (cid:96) such that B(cid:96) i ( x j , η (cid:96) ) (cid:54) = 0 and hence φ x j ( η (cid:96) ) (cid:54) = 0. Since (cid:82) S σ J i ( ω ) dω > ρ (2) ( M ( K i ) , φ x j ) ≥ − C K + (cid:82) S σ J i ( ω ) dω .Summarizing the computations, we have ρ (2) ( M ( K i ) , φ x j ) (cid:40) = 0 if x j = 0 , ≥ − C K + (cid:82) S σ J i ( ω ) dω if x j (cid:54) = 0 . Let d be the number of j such that x j (cid:54) = 0. Now we obtain that n (cid:88) j =1 ρ (2) ( M ( K i ) , φ x j ) ≥ d (cid:18) − C K + (cid:90) S σ J i ( ω ) dω (cid:19) , Since x (cid:54) = 0, we have d >
0. By Lemma 3.1(4), it follows that n (cid:88) j =1 ρ (2) ( M ( K i ) , φ x j ) > , which contradicts Equation (3.1). This proves Theorem 1.1(3). Proof of Theorem 1.1(2).
We show that K i are linearly independent in C , namely, nonontrivial linear combination of K i are slice. This can be easily shown by following thearguments in the proof of [Kim16, Theorem 4.2]. Moreover, a proof of Theorem 1.1(2) iseasier than that of [Kim16, Theorem 4.2] in the sense that it does not need the techni-calities used in the proof of [Kim16, Theorem 4.2] such as modules over noncommutativerings and the notion of algebraic n -solutions. For the reader’s convenience, we adapt theproof of [Kim16, Theorem 4.2] to our case, and give a proof of Theorem 1.1(2) below.Suppose to the contrary that L := i a i K i ( a i ∈ Z ), a nontrivial connected sum offinitely many copies of ± K i , is slice. We may assume a (cid:54) = 0 by reindexing, and bytaking the inverse of L if necessary we may assume further that a >
0. We constructa 4-manifold W by stacking up the following building blocks V , C , and V i . For a 4-manifold X and a homomorphism φ : π X → Γ where Γ is a group, for simplicity let S Γ ( X ) := sign (2)Γ ( X ) − sign( X ).(1) Let V be the exterior of a slice disk for L in D . Then, ∂V = M ( L ).(2) Let C be the standard cobordism between M ( L ) and (cid:96) a i M ( K i ) as constructedin the proof of Theorem 1.1(3). Turning C upside down, we may assume ∂C =( (cid:96) a i M ( K i )) (cid:96) ( − M ( L )).(3) For each i , let V i be the 4-manifold with ∂V i = M ( K i ) given by [Kim16, Lemma 4.1(1)]satisfying the following: suppose φ : π V i → Γ is a homomorphism where Γ is anamenable group lying in Strebel’s class D ( R ) for some ring R . Let d (cid:96) be the orderof φ ( η (cid:96) ) in Γ and let φ (cid:96) : π M ( J i ) → Z d (cid:96) be an epimorphism sending the meridianof J i to 1 ∈ Z d (cid:96) (where Z ∞ := Z ). Then, S Γ ( V i ) = (cid:80) m(cid:96) =1 ρ (2) ( M ( J i ) , φ (cid:96) ).(4) Let U be the 4-manifold with ∂U = M ( K ) (cid:96) ( − M ( K )) which is given by [Kim16,Lemma 4.1(2)] satisfying the following: suppose φ : π U → Γ is a homomorphismwhere Γ is a group as in (3). Let d (cid:96) and φ (cid:96) : π M ( J ) → Z d (cid:96) be as in (3). Then, S Γ ( U ) = − (cid:80) m(cid:96) =1 ρ (2) ( M ( J ) , φ (cid:96) ). NOTS HAVING THE SAME SEIFERT FORM 8
Let b := a − b i = | a i | for i ≥
2. For each i ≥
1, let V ri be a copy of − V i for1 ≤ r ≤ b i . Now we define W as follows: W := V (cid:91) ∂C − C (cid:91) ∂C + (cid:32) U (cid:97) (cid:32)(cid:97) i b i (cid:97) r =1 V ri (cid:33)(cid:33) where ∂C − := M ( L ) and ∂C + := (cid:96) a i M ( K i ). See Figure 1. Note that ∂W = M ( K ). VCU V ri V ri M ( L ) M ( K ) M ( K ) M ( K i ) M ( K i ) Figure 1.
Cobordism W Let Γ := π W/ ( π W ) (2) p as defined in Section 2 and φ : π W → Γ the projection. ByLemma 2.2, the group Γ is amenable and lies in Strebel’s class D ( Z p ). By abuse ofnotation, let φ also denote the restriction of φ to subspaces of W .From the definition of ρ (2) -invariants given in Section 2, we have ρ (2) ( M ( K ) , φ ) = S Γ ( W ). On the other hand, by Novikov additivity we have S Γ ( W ) = S Γ ( V ) + S Γ ( C ) + S Γ ( U ) + (cid:88) i b i (cid:88) r =1 S Γ ( V ri ) . We compute each term of the right-hand side of the above equation.(1) S Γ ( V ) = 0 by Theorem 2.1 since V is a slice disk exterior.(2) S Γ ( C ) = 0: since H ( C ) ∼ = H ( ∂ − C ), it follows that Coker { H ( ∂ − C ) → H ( C ) } =0. Now sign( C ) = sign (2)Γ ( C ) = 0 as we have seen in the proof of Theorem 1.1(3).(3) Let i > (cid:15) i := − a i / | a i | Then, S Γ ( V ri ) = (cid:15) i · (cid:80) m(cid:96) =1 ρ (2) ( M ( J i ) , φ (cid:96) ). Since η (cid:96) ∈ π ( M K ) (1) , we have φ ( η (cid:96) ) ∈ Γ (1) = π W (1) / ( π W ) (2) p . Since Γ (1) injects into( π W (1) /π W (2) ) ⊗ Z p , which is a Z p -vector space, we obtain that d (cid:96) = 0 or p .If d (cid:96) = 0, then φ (cid:96) is the trivial map and ρ (2) ( M ( J i ) , φ (cid:96) ) = 0 by (2.5) in [COT04,p.108]If d (cid:96) = p , then φ (cid:96) is a surjection to Z p , and by [CO12, Lemma 8.7] andLemma 3.1(3), we obtain ρ (2) ( M ( J i ) , φ (cid:96) ) = p (cid:80) p − r =0 σ J i ( e πr √− /p ) = 0.Similarly, S Γ ( V r ) = − ρ (2) ( M ( J ) , φ (cid:96) ) = 0 or − p (cid:80) p − r =0 σ J ( e πr √− /p ).Therefore, (cid:80) i (cid:80) b i r =1 S Γ ( V ri ) ≤ S Γ ( U ) = − (cid:80) m(cid:96) =1 ρ (2) ( M ( J ) , φ (cid:96) ), and similarly as in (3) above, ρ (2) ( M ( J ) , φ (cid:96) ) =0 if d (cid:96) = 0 and p (cid:80) p − r =0 σ J ( e πr √− /p ) if d (cid:96) = p . By Lemma 3.4 below, wecan conclude that S Γ ( U ) ≤ − p (cid:80) p − r =0 σ J ( e πr √− /p ). Lemma 3.4.
In (4) above, d (cid:96) = p for some (cid:96) . NOTS HAVING THE SAME SEIFERT FORM 9
Proof.
Let I be the image of the map i ∗ : H ( M ( K ); Z p [ t ± ]) → H ( W ; Z p [ t ± ]) where i ∗ is induced from the inclusion map. Then, rank Z p I ≥ rank Z p H ( M ( K ); Z p [ t ± ]).This can be seen by [Kim16, Theorem 5.2] observing that W is a (1)-cylinder. This is theonly place where we use the notion of ( n )-cylinders, and since the arguments for showing W is a (1)-cylinder is well-known for the experts, we give a brief proof that W is a (1)-cylinder below. One may refer to [CK08, Kim16] for the definition of an ( n )-cylinder, butwe will not use it below.By [Kim16, Lemma 4.1], the 4-manifolds V i and U are obtained as (1)-solutions anda (1)-cylinder, respectively. Since a (1)-solution is a (1)-cylinder (see [CK08, Proposi-tion 2.3]), V i are also (1)-cylinders. Since Coker { H ( ∂C ) → H ( C ) } = 0 and V is aslice disk exterior, the 4-manifolds C and V are also (1)-cylinders. Since W is a unionof (1)-cylinders along common boundary components, one can easily show that W is a(1)-cylinder following the arguments in the proof of [CK08, Proposition 2.6].Since p > a K by our choice of p , where a K is the top coefficient of ∆ K ( t ), we haverank Z p H ( M ( K ); Z p [ t ± ]) = deg ∆ K ( t ). Since ∆ K ( t ) is nontrivial, we have deg ∆ K ( t ) ≥
2. Therefore, rank Z p I ≥
1, and hence I (cid:54) = 0. Since the η (cid:96) generate H ( M ( K ); Z p [ t ± ]),this implies that i ∗ ( η (cid:96) ) (cid:54) = 0 in H ( W ; Z p [ t ± ]) for some (cid:96) . Since φ ( η (cid:96) ) ∈ Γ (1) and Γ (1) injects into ( π W (1) /π W (2) ) ⊗ Z p ∼ = H ( W ; Z p [ t ± ]), which is a Z p -vector space, itfollows that φ ( η (cid:96) ) has order 0 or p . But since i ∗ ( η (cid:96) ) (cid:54) = 0, we have φ ( η (cid:96) ) (cid:54) = 0. Therefore, φ ( η (cid:96) ) has order p . (cid:3) Now by (1)-(4) and Lemma 3.1(2), we conclude that ρ (2) ( M ( K ) , φ ) = S Γ ( W ) ≤ − p p − (cid:88) r =0 σ J ( e πr √− /p ) < − C K , which contradicts our choice of C K . This proves Theorem 1.1(2). Remark 3.5.
From the viewpoint of the solvable filtration {F n } in [COT03], for each i , K i ∈ F since η (cid:96) ∈ ( π M ( K )) (1) and J i ∈ F (a knot with zero Arf invariant lies in F ).Also, the proof for Theorem 1.1(2) is still available when V is a (1.5)-solution, and hencethe knots K i are, in fact, linearly independent in F / F . . References [Bao15] Yuanyuan Bao,
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