Heegaard Floer homology and several families of Brieskorn spheres

Topology and its Applications 160 (2013) 620–632

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Topology and its Applications www.elsevier.com/locate/topol

Heegaard Floer homology and several families of Brieskorn spheres Eamonn Tweedy Rice University, United States

a r t i c l e

i n f o

Article history: Received 15 October 2011 Received in revised form 7 January 2013 Accepted 7 January 2013 Keywords: Heegaard Floer homology Brieskorn homology sphere Seifert manifold 4-manifold 3-manifold Knot theory

a b s t r a c t In Ozsváth and Szabó (2003) [15] the authors gave a combinatorial description for the Heegaard Floer homology of boundaries of certain negative-definite plumbings. Némethi constructed a remarkable algorithm in Némethi (2005) [12] for executing these computations for almost-rational plumbings, and his work in Némethi (2007) [13] gives a formula computing the invariants for the Brieskorn homology spheres −Σ( p , q, pqn + 1). Here we give a formula for HF + (−Σ( p , q, pqn − 1)), generalizing the one for the n = 1 case given in Borodzik and Némethi (2011) [3]. We also compute HF + for the families −Σ(2, 5, k) and −Σ(2, 7, k). © 2013 Elsevier B.V. All rights reserved.

1. Introduction The Heegaard Floer homology package was first introduced by Ozsváth and Szabó [16], and has proven to be a useful collection of tools for the study of manifolds of dimensions 3 and 4. In particular, to a closed, oriented 3-manifold Y one can associate a graded Z[U ]-module HF + (Y ), which is the richest of the flavors (i.e. carries the most data). Combinatorial techniques and cut-and-paste techniques have since been developed to ease computation of various Heegaard Floer homologies  and those in [11] compute HF + ). (the methods in [7], [6], and [18] compute the graded Z-module HF However, Ozsváth and Szabó offered in [15] a combinatorial description of HF + for a 3-manifold which bounds a negative-definite plumbing graph. Némethi provided a nice combinatorial framework for this description in [12] via a very concrete algorithm: a plumbing graph leads to a function τ : Z0 → Z, which in turn induces an intermediate object known as a “graded root”; this gadget both determines HF + and carries some extra data relevant to singularity theory. For a very approachable “user’s guide” to Némethi’s algorithm, see [1]. The purpose of the present article is to write down formulae for the invariants HF + associated to some Brieskorn homology spheres; these include −Σ( p , q, pqn − 1), where p , q, n ∈ N with p , q coprime (Theorem 1) and −Σ(2, j , k) where j , k ∈ N, k is odd and coprime and j ∈ {5, 7} (Theorem 2). These computations attempt to further illustrate the usefulness of Némethi’s formula in allowing one to compute HF + for such infinite families of Seifert manifolds. Recall that

S 13/n ( T p ,q ) = −Σ( p , q, pqn − 1) and

3 S− 1/n ( T p ,q ) = Σ( p , q, pqn + 1),

where T p ,q denotes the right-handed ( p , q) torus knot and S r3 ( K ) denotes r-framed surgery on the knot K ⊂ S 3 . Negative Dehn surgeries on algebraic knots were extensively studied in [13], and Némethi writes down a closed formula 3 for HF + (− S − 1/n ( K )) in Section 5.6.2 of that paper. The torus knot T p ,q is indeed algebraic, and Némethi’s formula gives that when n > 0,

E-mail address: [email protected]. 0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2013.01.008

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Table 1 The structure of HF + for lots of Brieskorn spheres, as provided by Theorems 1 and 2 and Bordzik and Némethi’s Eq. (1).

∼ HF + HF + even red =

Manifold

d

T−+2 (1)⊕(n−1) ⊕

−Σ(2, 5, 10n − 1)

T0+ (1)⊕n ⊕

−Σ(2, 5, 10n + 1)

 n−1

T2i+ (1)⊕2

i =0

 n

T2i+ (1)⊕2

T0+ (1)⊕(n−1) ⊕

−Σ(2, 5, 10n − 3)

T−+2 (1)⊕n ⊕

−Σ(2, 5, 10n + 3)

 n−1

0

T−+4 (2)⊕(n−1) ⊕

−Σ(2, 7, 14n − 1)

T2i+ (1)⊕2

i =0

T2i+ (1)⊕2

i =0

 n−1

T−+2 (1)⊕(2n−2) ⊕

−Σ(2, 7, 14n − 3)

T0+ (1)⊕(2n+1) ⊕

−Σ(2, 7, 14n + 3)

T−+2 (1)⊕(2n−3) ⊕

−Σ(2, 7, 14n − 5)

T0+ (1)⊕(2n+2) ⊕

−Σ(2, 7, 14n + 5)

 n−1

i =0  n i =1  n−1 i =0  n i =1



+ 3 ⊕n HF + ⊕ even − S −1/n ( T p ,q ) = T0 (α g −1 )





3

0

 −2

   n−1 + ⊕2 T2i+ (1)⊕2 ⊕ T2n +4i (1)

i =1

+



−4

i =0 i =0     n n + ⊕2 T0+ (2)⊕(n) ⊕ T2i+ (1)⊕2 ⊕ T2n +4i (1)

−Σ(2, 7, 14n + 1)



−2



i =1

 n−1



HF odd − S −1/n ( T p ,q ) = 0,



n ( g −1 )



0

i =1

   n−1 + ⊕2 T2i+−2 (1)⊕2 ⊕ T2n +4i −2 (1) T2i+ (1)⊕2



i =0   n + ⊕2 ⊕ T2n +4i (1)



T2i+−2 (1)⊕2 ⊕

i =1  n−1 i =0

+ ⊕2 T2n +4i −2 (1)

−2 0



   n + ⊕2 T2i+ (1)⊕2 ⊕ T2n +4i (1)

−2 0

i =1

⊕2 Tc+ (n,i ) (α g −1+i /n ) ,

i =1



3

and d − S −1/n ( T p ,q ) = 0

(1)

where c (n, i ) := ( i /n + 1)({i /n}n + i ). Note that {x} := x − x and the α j are non-negative integers defined in Section 2.2. The Z[U ]-module Tm+ (k) is a copy of the quotient module Z[U ]/U k Z[U ] graded such that U (k−1) has grading m and such that the U -action has degree −2; Section 2.4 below elaborates on this object and others arising in Heegaard Floer theory. The case of +1-surgery on T p ,q was studied by Borodzik and Némethi in [3], and a formula for HF + was given there. Presently, we extend that computation to +1/n-surgery for n > 0 to provide the following formula. Theorem 1. Let p , q > 0 be coprime integers, and let T p ,q denote the torus right-handed ( p , q) torus knot. Then +





HF even S 13/n ( T p ,q )





= T−+2α g −1 (α g −1 )⊕(n−1)

HF + S 13/n ( T p ,q ) = 0, odd



n ( g −1 )

⊕ 



Td+(n,i )−2α

i =1

g −1+i /n

(α g −1+i /n )⊕2 ,

and d S 13/n ( T p ,q ) = −2α g −1

where d(n, i ) := (i /n)({(i − 1)/n}n + i − 1). Némethi’s method doesn’t rely on the surgery presentation given above, and may be applied to many other infinite families of Seifert manifolds. For the sake of illustration, we compute HF + for all Brieskorn homology spheres of the form −Σ(2, 5, k) or −Σ(2, 7, k). This computation is provided by the following (along with Eq. (1) and Theorem 1), which is proven in Section 4. The main technical input comes from Lemma 13, which is stated and proved in Appendix A. Theorem 2. Fix n ∈ N, and let M be any of the Brieskorn homology spheres −Σ(2, 5, 10n ± 3), −Σ(2, 7, 14n ± 3), or −Σ(2, 7, 14n ± 5). Then HF + ( M ) is as characterized by Table 1. Let p be a prime and K ⊂ S 3 be a knot. Using a particular d-invariant for the pn -fold cover of S 3 branched along K , Manolescu and Owens [9] (for pn = 2) and Jabuka [5] (for any prime p and any n ∈ Z>0 ) define a concordance invariant δ pn ∈ Z; see Section 2.7 for the definition of this invariant. Theorem 2 provides some new δ pn -invariants for torus knots (although a few of the examples below appeared in [5]). Recall from [8] that when p , q, r > 0 are pairwise coprime, in fact

−Σ( p , q, r ) = Σ p ( T q,r ) = Σq ( T p ,r ) = Σr ( T p ,q ).

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Corollary 3. For p , q ∈ N coprime, let T p ,q denote the right-handed ( p , q)-torus knot. (i) Let k = 10n ± 3, where n ∈ N. Then



δ2 ( T 5,k ) = δ5 ( T 2,k ) =

−4, k = 10n + 3, k = 10n − 3.

0,

Moreover, whenever k is a prime power,

 δk ( T 2,5 ) =

−4, k = 10n + 3 (e.g. n = 1, 2, 4, 5, 7, . . .), k = 10n − 3 (e.g. n = 1, 2, 4, 5, 7, . . .).

0,

(ii) Let k = 14n ± 3 or k = 14n ± 5, where n ∈ N. Then



δ2 ( T 7,k ) = δ7 ( T 2,k ) =

−4, k = 14n − 3 or k = 14n − 5, k = 14n + 3 or k = 14n + 5.

0,

Moreover, whenever k is a prime power,

⎧ −4, ⎪ ⎪ ⎪ ⎨ 0, δk ( T 2,7 ) = ⎪ −4, ⎪ ⎪ ⎩ 0,

k = 14n − 3 (e.g. n = 1, 4, 5, 8, 10, . . .), k = 14n + 3 (e.g. n = 1, 2, 4, 5, 7, . . .), k = 14n + 5 (e.g. n = 1, 3, 4, 6, 7, . . .), k = 14n − 5 (e.g. n = 2, 3, 6, 8, 11, . . .).

Remark 4. The families T 5,k and T 7,k provide several new infinite families of (non-alternating, of course) knots such that −δ2 = σ /2 (cf. [9]). For the reader’s convenience, we list those values here: Knot K

T 5,10n+3

T 5,10n−3

T 7,14n+3

T 7,14n−3

T 7,14n+5

T 7,14n−5

σ ( K )/2

4(3n + 1) 4

4(3n − 1) 0

4(6n + 1) 0

4(6n − 1) 4

4(6n + 2) 0

4(6n − 2) 4

−δ2 ( K )

Conjecture 5. Let p > 1 be an odd integer and let k be an integer with gcd(k, 2p ) = 1 and k ≡ ±1 (mod 2p ), then

  0, d −Σ(2, p , 2pn − k) = −2,    0, d −Σ(2, p , 2pn + k) = −2, 

p ≡ 1 (mod 4), p ≡ 3 (mod 4)

and

p ≡ 3 (mod 4), p ≡ 1 (mod 4).

Remark 6. Along with Theorem 1 and Eq. (1), Conjecture 5 would determine the invariant δ2 for all torus knots T p ,q with p , q odd. 2. Preliminaries 2.1. Seifert fibered integer homology spheres Recall that the Seifert fibered space Σ := Σ(e 0 , (a1 , b1 ), (a2 , b2 ), . . . , (am , bm )) bounds a plumbed negative-definite 4manifold whose plumbing graph is star-shaped, consisting of m “arms” emanating from a central vertex. The central vertex is labeled with weight e 0 , and the ith arm is a chain of ni vertices with labels −k1 , −k2 , . . . , −kni (ordered outward from the center), where

ai bi

= k1 −

1 k2 −

.

1

..

.−

1 kni

Now define the quantities

e := e 0 +

m

bi i =1

ai

and

ε :=

1 e

 2−m+

m

1 i =1

ai

 .

E. Tweedy / Topology and its Applications 160 (2013) 620–632

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If we assume e < 0, then Σ is an integer homology sphere if and only if e = −1/(a1 a2 . . . am ), i.e.

  m

a1 a2 . . . am . −1 = e 0 a1 a2 . . . am + bi ai

i =1

This equation implies that the residue of b i modulo ai is determined by the a j ’s. 2.2. Torus knots We review some notation related to the torus knot T p ,q which appears in some formulae in the introduction. Let S p ,q ⊂

Z0 denote the semigroup

  S p ,q := ap + bq: (a, b) ∈ Z20 . Now Z0 \ S p ,q is finite, and in fact

|Z0 \ S p ,q | =

( p − 1)(q − 1) 2

= g 3 ( T p ,q ) =: g ,

For i  0, we define a sequence of numbers

the 3-genus of T p ,q .

αi ∈ Z0 via

αi := #{s ∈/ S p,q : s > i }. One can verify that in fact g = α0  α1  · · ·  α2g −3  α2g −2 = 1 and

αi = 0 for i > 2g − 2.

2.3. Dedekind sums Recall the “sawtooth function” · : R → R, where



x :=

0,

x ∈ Z,

x − x − 12 ,

x∈ / Z.

For h, k ∈ Z \ {0}, one can define the classical Dedekind sum

s(h, k) :=

 k−1  

i hi i =1

k

k

.

We’ll make use of a particular formula found in [2] involving the Euclidean algorithm. Assume that 0 < h < k and that r0 , r1 , . . . , rn+1 are the remainders obtained when the Euclidean algorithm is applied to h and k, i.e.

r0 := k,

r1 := h,

r j +1 ≡ r j −1

(mod r j ) (with 1  r j +1 < r j ),

and rn+1 = 1.

Then in fact the Dedekind sum can be computed via

s(h, k) =

1 12

 n +1  

1 + r 2j + r 2j −1 1 + (−1)n j +1 − (−1) . r j r j −1

j =1

(2)

8

2.4. Heegaard Floer homology Let Y be a rational homology 3-sphere, and fix s ∈ Spinc (Y ). We study the Q-graded Heegaard Floer homology groups HF + (Y , s), defined by Ozsváth and Szabó in [16] and [17]. Define the graded Z[U ]-modules

T + :=

Z[U , U −1 ] U · Z[U ]

and

T + (n) :=

Z U (−n+1) , U (−n+2) , . . . , U · Z[U ]





where deg U k = −2k.

More generally, given a graded Z[U ]-module M with k-homogeneous elements M k and some d ∈ Q, let M [d] denote the graded Z[U ] module with M [d]k+d = M k . Then define the shifted modules Td+ := T + [d], Td+ (n) := T + (n)[d]. Recall that the Q-graded Heegaard Floer groups decompose as

HF + (Y , s) ∼ (Y , s), = Td+(Y ,s) ⊕ HF + red where the first summand is the image of the projection map HF ∞ (Y , s) → HF + (Y , s) and second is its quotient. Note that the invariant d(Y , s) ∈ Q is the so-called correction term or d-invariant associated to the pair (Y , s), first introduced in [14].

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E. Tweedy / Topology and its Applications 160 (2013) 620–632

Note that when Y is an integer homology sphere, there is only one element in Spinc (Y ); in this case, suppress the “s” and just write d(Y ). Recall also that HF + is relatively Z-graded and carries a well-defined absolute Z/2Z-grading. With respect to this gradfurther decomposes as ing, HF + red

HF + (Y ) = HF + (Y ) ⊕ HF + even ( Y ). red odd Remark 7. Given HF + (−Y ), it is straightforward to compute HF + (Y ). Indeed, one should first use the long exact sequence

· · · −→ HF − (−Y ) −→ HF ∞ (−Y ) −→ HF + (−Y ) −→ · · · to recover HF − (−Y ), and then use the fact that H F ∗+ (Y ) ∼ = H F−

(−∗−2)

(−Y ).

2.5. Némethi’s algorithm In [12], Némethi describes a procedure for computing the Heegaard Floer homology for boundaries of negative-definite almost-rational plumbings. A plumbing is almost-rational if its graph contains at most one bad vertex, i.e. a vertex v with degree( v ) > |weight( v )|. Note that the star-shaped plumbing graph associated to a Seifert manifold can always be drawn such that no vertices are bad except possibly the central one. We’ll briefly describe the algorithm and then review in detail the step which is particularly relevant in this paper; see [1] for a more complete user’s guide. Let Γ be the plumbing graph and let X (Γ ) denote the associated plumbed 4-manifold. The algorithm uses Γ to induce a computation sequence of vectors in H 2 ( X (Γ )), which in turn provides a tau function τ : Z0 → Z. The sequence (τ (i )) is eventually non-decreasing. One constructs a reduced version τ˜ of the function by throwing out all repetition in the sequence (τ (i )), keeping only the local extrema, and re-indexing the result (which is necessarily a finite sequence of integers). The function τ˜ generates a Z-graded infinite tree called a graded root; this tree has an obvious Z[U ]-action and recovers the module HF + (−∂ X (Γ )). The process of constructing the graded root from the function τ˜ and subsequently recovering HF + (−∂ X (Γ )) will be reviewed below. Further analysis of the function τ , as well as a notion of complexity of a graded root, can be found in [4]. Suppose that one has generated a reduced tau function, viewed as a sequence of integers τ˜ = (t 0 , t 1 , . . . , t 2k ). Following the notation from [1], let R i denote the infinite oriented tree with vertex set V = { v j : j  i } and edge set E = {[ v j → v ( j +1) ]: j  i }, graded by the function χ : V → Z with χ ( v j ) = j. Recall that the local minima of the sequence τ˜ are t 0 , t 2 , . . . , t 2k , and that for 1  j  k, there are grading-preserving injections of R t(2 j−1) into R t(2 j−2) and into R t2 j Then construct a new infinite oriented tree R τ = (Vτ , Eτ ) via

R τ := ( R t0  R t1  · · ·  R t2k )/ ∼ where the equivalence relation ∼ identifies R t(2 j−1) with its images in both R t(2 j−2) and R t2 j for 1  j  k. The tree R τ inherits a grading function χτ from the trees R t i , and the pair ( R τ , χτ ) is called the graded root associated to τ . The vertices in the tree represent generators of the module HF + (−∂ X (Γ )), and the U -action is encoded the oriented edges. Now define the set of functions

  H( R τ , χτ ) := φ : V → T(+ 0) : U · φ( v ) = φ( w ) if [ v → w ] ∈ Eτ . This admits a natural Z[U ]-module structure, and can be graded by the following rule: φ is a homogeneous element of degree d if and only if for each v ∈ Vτ , φ( v ) ∈ T(+ 0) is a homogeneous element of degree d − χτ ( v ). It is shown in [12] that

in fact H( R τ , χτ ) ∼ = HF + (−∂ X (Γ )) as relatively-graded Z[U ]-modules. To recover the absolute grading on HF + (−∂ X (Γ )), one may use the correction term d(−∂ X (Γ )); this integer can be computed using formula (5) or via other methods. In practice, one can write down HF + (−∂ X (Γ )) as a direct sum, with one summand for each “leaf” (i.e. vertex of degree 1) of the graded root. Let V0 ⊂ Vτ denote the set of leaves, let d := d(−∂ X (Γ )), and let

G := min





χτ ( v ): v ∈ V0 .

We then describe the notion of the length λ( v ) of a leaf v ∈ V0 . If v ∈ V0 , then v coincides with the lowest-grading vertex of some R t2i . Then define

λ( v ) := min(t (2i +1) − t 2i , t (2i −1) − t 2i ) > 0. Fix some v 0 ∈ V0 with χτ ( v 0 ) = G. The leaf v 0 provides the summand Td+ , and each other leaf provides summand of HF + (−∂ X (Γ )). We then have that red





HF + −∂ X (Γ ) = Td+ ⊕



v ∈V0 v = v 0

   Td++2(χτ ( v )−G ) λ( v ) .

E. Tweedy / Topology and its Applications 160 (2013) 620–632

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Fig. 1. The graded trees R t i determined by the sequence τ˜ for Σ(2, 5, 13) are shown on the left, and the resulting graded root ( R τ , χτ ) is shown on the right. Black dots represent vertices and the scale on the far left indicates the grading.

For example, using the results of Table 2 below, the reduced sequence for Σ(2, 5, 13) is

τ˜ = (0, 1, −1, 0, −1, 1, 0) Fig. 1 exhibits the graded root induced by this sequence. According to Table 1, d(−Σ(2, 5, 13)) = −2 – along with Fig. 1, this implies that

 HF

+

 −Σ(2, 5, 13) ∼ = T−+2 ⊕ T−+2 (1) ⊕ T0+ (1)⊕2 .

Remark 8. Given a particular plumbing graph, one would use the reduced function τ˜ to draw the graded root in practice; however, notice that Eq. (3) indeed involves the full tau function τ , and that’s the one commonly used in the arguments here. 2.6. Some formulas of Bordzik and Némethi Let Σ be a Seifert fibered homology sphere (as in Section 2.1). In [3], Borodzik and Némethi characterize the d-invariant and the tau function for Σ in terms of its Seifert invariants (e 0 , (a1 , b1 ), . . . , (am , bm )). Proposition 2.2 of [3] states that for each k ∈ Z0 ,

τ (k) =

k −1

 j,

where  j := 1 − je 0 −

j =0

 m 

jb i . i =1

ai

(3)

There is an alternate form for  j which is sometimes more useful. For any b ∈ Z and a1 , . . . , am ∈ Z>0 , let

εa (b) :=

m



εai (b), where εai (b) :=

i =1

1, ai |b, 0,

else.

Then we have that

j = 1 −

m 2

+

j a1 · · · am

+

εa ( j ) 2

 m 

jb i . + i =1

(4)

ai

The d-invariant is then given by

d(Σ) =

1 4

 2

ε e + e + 5 − 12

m

i =1

 s(b i , ai ) − 2 min τ (k). k0

(5)

2.7. The concordance invariant δ pn Let K ⊂ S 3 be a knot, let p be a prime, and let n ∈ N. Then let Σ pn ( K ) denote the pn -fold branched cover of S 3 branched along the knot K . Σ pn ( K ) is a rational homology sphere, and we let s0 ∈ Spinc (Σ pn ( K )) denote the element induced by the unique spin-structure. Manolescu and Owens [9] (for pn = 2) and Jabuka [5] (for general pn ) define

  δ pn ( K ) := 2d Σ pn ( K ), s0 ∈ Z. This number is an invariant of the smooth knot concordance class of K , and in fact provides a homomorphism from the smooth knot concordance group to Z. Corollary 3 above provides some new computations of δ pn for some torus knots.

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E. Tweedy / Topology and its Applications 160 (2013) 620–632

3. The Brieskorn spheres Σ( p , q, pqn − 1) Let p , q > 0 be coprime integers. Recall that the Brieskorn homology sphere Σ( p , q, pqn − 1) is a Seifert fibered space with Seifert invariants (e 0 , ( p , p  ), (q, q ), (r , r  )), where e 0 = −2, r = pqn − 1, r  = pqn − n − 1, and p  , q uniquely determined by the restrictions

0 < p < p,

0 < q < q,

pq ≡ 1 (mod q),

and qp  ≡ 1 (mod p ).

Remark 9. When p = 2, the above constraints imply that p  = 1 and q = (q + 1)/2. The reader can verify that these parameters lead to the plumbing graph found in Fig. 2(b) in Appendix A. Note that we don’t need the graphs to compute the Heegaard Floer groups, but rather only the Seifert invariants. Némethi gives a formula for the function τ for a Seifert manifold in [12], and in [3] uses it to compute HF + for +1surgery on T p ,q . In order to extend his result to +1/n-surgery (n ∈ N), we first characterize the function τ for this case. Lemma 10. Let S p ,q denote the semigroup of Z0 generated by p and q. Additionally, let N := n(2g − 1). Then the following hold. (i) The function τ : Z0 → Z attains its local maxima (resp. minima) at the points M i (resp. mi ), where

M i := pqi + 1

(for 0  i  N − 2) and mi := pqi − i /n (for 0  i  N − 1).

(ii) For 0  i  N − 2,



 

τ ( M i ) − τ (mi ) = # s ∈ S p,q : s  

i

> 0 and

n



τ ( M i ) − τ (mi+1 ) = # s ∈/ S p,q : s 

i+1 n



(6)

 + 1 = α i+1 > 0.

(7)

n

(iii) The sequence (mi ) satisfies

⎧ n  0 for i ∈ {0, . . . , N − − 1}, ⎪ 2 ⎨ N −n N +n τ (mi+1 ) − τ (mi ) = 0 for i ∈ { 2 , . . . , 2 − 2}, ⎪ ⎩ n  0 for i ∈ { N + − 1, . . . , N − 2} 2 n n and thus τ achieves its global minimum value at the points mi with i ∈ { N − , . . . , N+ − 1}. 2 2

Proof of (i). For each i ∈ Z0 , define the numbers M i and mi via the expressions stated in the lemma (we’ll show that τ attains its local extrema at some of these points, i.e. the ones with indices restricted as in the lemma). To this end, fix j ∈ Z0 and compute  j := τ ( j + 1) − τ ( j ). We’ll first assume that M i  j < M i +1 , and employ an analysis similar to that in [3]. In particular, recall that for any integer s ∈ [0, pq),

s ∈ S p ,q

⇔

s = α p + βq

for some 0  α < q, 0  β < p

/ S p ,q s∈

⇔

s + pq = α p + β q

and

for some 0  α < q, 0  β < p .

Following [3], one can use this fact along with Eq. (4) to show that

j =

⎧ ⎨

jn

−i pqn−1

when (i + 1) pq − j ∈ S p ,q ,

⎩

jn

−i pqn−1

− 1 when (i + 1) pq − j ∈ / S p ,q .

First assume that 0  i  N − 2. In this case, whenever (i + 1) pq − j ∈ S p ,q , one finds that



jn



 −i=

pqn − 1

0

when M i  j < mi +1 ,

(8)

1 when mi +1  j < M i +1 .

On the other hand, whenever (i + 1) pq − j ∈ / S p ,q ,



jn pqn − 1



 −i−1=

−1 when M i  j < mi +1 , 0

(9)

when mi +1  j < M i +1 .

Now when i  N − 1, we find that  j  0 regardless of whether (i + 1) pq − j ∈ S p ,q .

2

E. Tweedy / Topology and its Applications 160 (2013) 620–632

627

Proof of (ii). Eqs. (6) and (7) follow from Eqs. (8) and (9), bearing in mind that

 

mi  j  M i

⇔

 M i  j  m i +1

i

pq − 1  (i + 1) pq − j  pq +

⇔

i+1



n

and

n

+ 1  (i + 1) pq − j  pq − 1.

The quantities in Eqs. (6) and (7) are strictly positive because 2g − 1 ∈ / S p ,q and 0 ∈ S p ,q .

2

Proof of (iii). Observe that

k ∈ S p ,q

⇔

2g − 1 − k ∈ / S p ,q .

(10)

Along with Eqs. (6) and (7), this implies that



 

τ (mi+1 ) − τ (mi ) = # k ∈/ S p,q : k  2g − 1 − and the result follows.

i

n

    i+1 −# k∈ +1 , / S p ,q : k  n

2

Notice that part (iii) of Lemma 10 implies that the length of the ith leaf is equal to



λi :=

τ ( M i ) − τ (mi )

if 0  i 

τ ( M i−1 ) − τ (mi ) if i >

N +n 2

N +n 2

− 2,

− 2.

In fact, the graded root determined by the function

τ is highly symmetric; the following makes this more precise.

Lemma 11. Let nk  i < nk + n for some 0  k  g − 2. (i) Lengths of leaves are symmetric, i.e.

τ ( Mn( g −1)−(i+1) ) − τ (mn( g −1)−(i+1) ) = α g +k = τ ( Mng +(i−1) ) − τ (mng +i ). (ii) Grading levels of leaves are symmetric, i.e.

2τ (mn( g −1)−(i +1) ) = (k + 1)(2i − nk) − 2α g +k + C (n, g ) = 2τ (mng +i ), where C (n, g ) = g (n − ng + 2). Moreover, there is a “bunch” of n leaves at the minimum grading level, i.e. for n( g − 1)  i  ng − 2,

τ ( M i ) − τ (mi ) = α g −1 and 2τ (mi ) = −2α g −1 + C (n, g ). Proof of (i). Notice that since nk  i < n(k + 1),

g −k−2

n ( g − 1) − ( i + 1) n



< g − k − 1,

and so

n ( g − 1) − ( i + 1)



n

= g − k − 2.

Now by Eqs. (6) and (10),

τ ( Mn( g −1)−(i+1) ) − τ (mn( g −1)−(i+1) ) = #{s ∈ S p,q : s  g − k − 2} = α g +k . The second equality of part (i) follows similarly, using Eq. (7). Proof of (ii). We have that



τ ( M 0 ) = 1 and τ ( M i+1 ) − τ ( M i ) = and so

τ (M i ) = 1 +

i  

m m =1

n

i+1

2

 + 1 − g for 0  i  N − 3,

n

 +1− g

for 1  i  N − 2.

Now we let i = nk + j, where 0  j < n. Then 0  n − ( j + 1) < n and

n( g − 1) − (i + 1) = n( g − 2 − k) + n − ( j + 1).

(11)

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E. Tweedy / Topology and its Applications 160 (2013) 620–632

As a result, we have that n( g −1)−(i +1)

m

 =

n

m =1

=

n( g − 3 − k)( g − 2 − k) 2

+ (n − j )( g − 2 − k)

 n( g − 2 − k )  n( g − 1 − k ) − 2 j . 2

Along with Eq. (11), this implies that

2τ ( M n( g −1)−(i +1) ) = 2 + ( g − 2 − k)(n − gn − nk − 2 j ) + 2(n − j − 1)(1 − g )

= (k + 1)(nk + 2 j ) + g (n − gn + 2) = (k + 1)(2i − nk) + C (n, g ). The first equality of part (ii) now follows from the first equality of part (i) of this lemma. The second equality of part (ii) can be proved similarly. 2 Proof of Theorem 1. For 0  i  n( g − 1), let G i be given by



   (i − 1)/n n + i − 1 − 2α g −1+i /n + C (n, g ).

G i := i /n

Lemmas 10 and 11 describe the structure of the graded root. Via the process described in the second half of Section 2.5, one can infer that for some S ∈ Z,





HF + S 13/n ( T p ,q ) = TG+0 + S (α g −1 )⊕(n−1) ⊕ red



n ( g −1 )



 i =1

TG+i + S (α g −1+i /n )⊕2 and

d S 13/n ( T p ,q ) = −2α g −1 + C (n, g ) + S . We claim that in fact S + C (n, g ) = 0, which would finish the proof. Recall that Moser showed in [10] that S 3pq−1 ( T p ,q ) is a lens space. In [14], Ozsváth and Szabó computed d-invariants for surgeries on lens space knots (knots in S 3 for which there exist positive integer surgeries yielding lens spaces); in particular, for n > 0, the d-invariant of 1/n surgery on a lens space knot is independent of n. Along with Eq. (1), this implies that









d S 13/n ( T p ,q ) = d S 13 ( T p ,q ) = −2α g −1 .

2

Remark 12. One could alternatively use Eqs. (2) and (5) to compute the above d-invariant. 4. The Brieskorn spheres Σ(2, 5, k) and Σ(2, 7, k) We’ll compute HF + for these manifolds using the formulae in Section 2.6, and the only inputs we’ll need are the Seifert invariants (though the interested reader can see Figs. 3(a)–3(f) in Appendix A for associated plumbing graphs). The discussion in Section 2.1 tells us that we can write

  Σ(2, 5, 10n − 3) = Σ −1, (2, 1), (5, 1), (10n − 3, 3n − 1) ,   Σ(2, 5, 10n + 3) = Σ −2, (2, 1), (5, 4), (10n + 3, 7n + 2) ,   Σ(2, 7, 14n − 5) = Σ −2, (2, 1), (7, 5), (14n − 5, 11n − 4) ,   Σ(2, 7, 14n − 3) = Σ −2, (2, 1), (7, 6), (14n − 3, 9n − 2) ,   Σ(2, 7, 14n + 3) = Σ −1, (2, 1), (7, 1), (14n + 3, 5n + 1) ,   Σ(2, 7, 14n + 5) = Σ −1, (2, 1), (7, 2), (14n + 5, 3n + 1) . The structure of HF + can be inferred from the graded root, which is specified by the tau function. We state and prove Lemma 13 in Appendix A, which characterizes τ for Σ(2, 5, k) and Σ(2, 7, k). With those results in mind, we have the necessary ingredients for proving Theorem 2. Proof of Theorem 2. We prove the statement for −Σ(2, 7, 14n + 3), and leave the proofs for the other five families as exercises – all of the arguments are quite similar. First notice in Table 2 that the local extrema of the tau function occur symmetrically, with

E. Tweedy / Topology and its Applications 160 (2013) 620–632

629

Table 2 Features of the tau functions for the manifolds Σ(2, 5, 10n ± 3). Cases marked with “†” only appear when n > 1. Manifold

Σ(2, 5, 10n − 3)

Mi

10i + 1, i ∈ [0, 3n − 2]

 mi



τ ( M i ) − τ (mi ) 

τ ( M i ) − τ (mi +1 ) 

τ (M i ) 

τ (mi )



τ (m6n+1−i ) = τ (mi ) = τ ( M 6n−i ) = τ ( M i ) =

0, 10i − 2, 10i − 8,

i=0 i ∈ [1, n − 1]† i ∈ [n, 3n − 1]

1, 2,

i ∈ [0, 2n − 1] i ∈ [2n, 3n − 2]†

2, 1,

i ∈ [0, n − 2]† i ∈ [n − 1, 3n − 2]

1 − i, 2 − n, 3 − 3n + i ,

i ∈ [0, n − 1] i ∈ [n, 2n − 2]† i ∈ [2n − 1, 3n − 2]

−i , i ∈ [0, n − 1] 2 − n, i ∈ [n, 2n − 1] 3 − 3n + i , i ∈ [2n, 3n − 1]

−2i , i ∈ [0, n],

−n − i , i ∈ [n + 1, 2n]  1 − 2i , i ∈ [0, n],

Σ(2, 5, 10n + 3)  10i + 1, i ∈ [0, 2n − 1] 10i + 7, i ∈ [2n, 3n − 1] 10i , i ∈ [0, 3n]

  



1, 2,

i ∈ [0, 2n − 1] i ∈ [2n, 3n − 1]

2, 1,

i ∈ [0, n − 1] i ∈ [n, 3n − 1]

1 − i, 1 − n, 2 − 3n + i ,

i ∈ [0, n − 1] i ∈ [n, 2n − 1] i ∈ [2n, 3n − 1]

i ∈ [0, n − 1] −i , i ∈ [n, 2n] −n, −3n + i , i ∈ [2n + 1, 3n]

and

1 − n − ii ∈ [n + 1, 2n].

Lemma 13, along with this observation, implies that for some shift S ∈ Z, +

HF red

 n   n     + + + ⊕(2n−1) ⊕2 ⊕2 −Σ(2, 7, 14n + 3) = T−6n+ S (1) ⊕ T−6n+2i + S (1) T−4n+4i + S (1) ⊕ ,





i =1



and

i =1

d −Σ(2, 7, 14n + 3) = −6n + S . Unfortunately, these manifolds aren’t surgeries on torus knots. Fortunately, it is straightforward to compute the d-invariants directly via Eq. (5). Applying the Euclidean algorithm to k = 14n + 3 and h = 5n + 1, one obtains the remainder sequences

(14n + 3, 5n + 1, 4n + 1, n, n − 1, 1) for n > 2, With these in hand, we find that







d −Σ(2, 7, 14n + 3) = −

1 4

(31, 11, 9, 2, 1) for n = 2,

 (−24n) − 2(−3n) = 0 and so S = 6n.

and

(17, 6, 5, 1) for n = 1.

2

Acknowledgements I would like to thank Ça˘gri Karakurt for introducing me to Némethi’s graded roots during his visit to Rice in Spring 2012 (and he and his coauthor for the useful primer in [1]). I would also like to thank Tye Lidman for some helpful conversations. Appendix A. Tau functions used to prove Theorem 2 Lemma 13. Fix n ∈ N. For the tau function of Σ(2, 5, 10n ± 3) (resp. Σ(2, 7, 14n ± 3), resp. Σ(2, 7, 14n ± 5)), the following hold. (i) The function τ : Z0 → Z attains its local maxima at the points M i and local minima at the points mi , where these sequences are defined in Table 2 (resp. Table 3, resp. Table 4). (ii) Changes in τ between consecutive extrema and the values of the function at these extrema are as given in Table 2 (resp 3, resp. 4). Proof of (i). We give the proof of the lemma for Σ(2, 7, 14n + 3) and leave the arguments for the other five families as exercises (as they work analogously). As in the proof of Lemma 10, first define the sequences M i and mi via the expressions given in Table 2 (we also define M 6n+1 := 70n + 15, although this won’t end up being the location of a local extremum). We’ll then compute  j = τ ( j + 1) − τ ( j ) in each of four cases: Case 1. Let M i  j < M i +1 , where 0  i  2n − 1. Now we can write j = 14i + 1 + k, where 0  k  13. Eq. (3) indicates that

630

E. Tweedy / Topology and its Applications 160 (2013) 620–632

Table 3 Features of the tau functions for the manifolds Σ(2, 7, 14n ± 3). Cases marked with “†” only appear when n > 1.

Σ(2, 7, 14n − 3) ⎧ 14i + 1, ⎪ ⎪ ⎨ 14(n + i ) − 5

Manifold

2 i −1 )+1 2

Mi

Σ(2, 7, 14n + 3) ⎧ 14i + 1, ⎪ ⎪ ⎨ 14(n + i ) + 1

i ∈ [0, 2n − 1] i ∈ [2n, 4n − 2], i even

2 i −1 )+7 2

⎪ 14(n + ⎪ ⎩ 14(i − n) + 1, ⎧ 14i , ⎪ ⎪ ⎨ 14(n + i )

⎪ 14(n + i ∈ [2n, 4n − 2], i odd ⎪ ⎩ 14(i − n) + 9, i ∈ [4n − 1, 6n − 3] ⎧ 14i , i ∈ [0, 2n − 1] ⎪ ⎪ ⎨ 14(n + i ) − 8 i ∈ [2n, 4n − 1], i even 2 i − 1 ⎪ i ∈ [2n, 4n − 1], i odd ⎪ 14(n + 2 ) ⎩ i ∈ [4n, 6n − 2] 14(i − n),  1, i ∈ [0, 4n − 2] 2, i ∈ [4n − 1, 5n − 2] 3, i ∈ [5n − 1, 6n − 3]† ⎧ ⎨ 3, i ∈ [0, n − 2]† 2, i ∈ [n − 1, 2n − 2] ⎩ 1, i ∈ [2n − 1, 6n − 3] ⎧ 1 − 2i , i ∈ [0, n − 1] ⎪ ⎪ ⎪ i ∈ [n, 2n − 1] ⎨ 2 − n − i, 3 − 3n, i ∈ [2n, 4n − 3]† ⎪ ⎪ i ∈ [4n − 2, 5n − 2] ⎪ ⎩ 5 − 7n + i , 7 − 12n + 2i , i ∈ [5n − 1, 6n − 3]† ⎧ −2i , i ∈ [0, n − 1] ⎪ ⎪ ⎪ 1 − n − i, i ∈ [n, 2n − 1] ⎨ 2 − 3n, i ∈ [2n, 4n − 2] ⎪ ⎪ i ∈ [4n − 1, 5n − 2] ⎪ 3 − 7n + i , ⎩ 4 − 12n + 2i , i ∈ [5n − 1, 6n − 2]

mi

τ ( M i ) − τ (mi )

τ ( M i ) − τ (mi +1 )

τ (M i )

τ (mi )

2 i −1 )+6 2

i ∈ [0, 2n] i ∈ [2n + 1, 4n], i even i ∈ [2n + 1, 4n], i odd i ∈ [4n + 1, 6n] i ∈ [0, 2n] i ∈ [2n + 1, 4n + 1], i even

⎪ 14(n + i ∈ [2n + 1, 4n + 1], i odd ⎪ ⎩ i ∈ [4n + 2, 6n + 1] 14(i − n) − 8,  1, i ∈ [0, 4n] 2, i ∈ [4n + 1, 5n] 3, i ∈ [5n + 1, 6n] 

3, 2, 1,

i ∈ [0, n − 1] i ∈ [n, 2n − 1] i ∈ [2n, 6n]

⎧ 1 − 2i , i ∈ [0, n] ⎪ ⎪ ⎪ i ∈ [n + 1, 2n − 1] ⎨ 1 − n − i, 1 − 3n, i ∈ [2n, 4n] ⎪ ⎪ i ∈ [4n + 1, 5n] ⎪ ⎩ 1 − 7n + i , 1 − 12n + 2i , i ∈ [5n + 1, 6n] ⎧ −2i , i ∈ [0, n] ⎪ ⎪ ⎪ −n − i , i ∈ [n + 1, 2n − 1] ⎨ i ∈ [2n, 4n + 1] −3n, ⎪ ⎪ i ∈ [4n + 2, 5n] ⎪ −1 − 7n + i , ⎩ −2 − 12n + 2i , i ∈ [5n + 1, 6n + 1]

Table 4 Features of the tau functions for the manifolds Σ(2, 7, 14n ± 5). Cases marked with “†” only appear when n > 1.

Σ(2, 7, 14n − 5) ⎧ 14i + 1, i ∈ [0, 2n − 1] ⎪ ⎪ ⎨ 14(n + i ) − 9 i ∈ [2n, 4n − 3], i even† 2 † i −1 ⎪ ⎪ 14(n + 2 ) + 1 i ∈ [2n, 4n − 3], i odd ⎩ 14(i − n) + 15, i ∈ [4n − 2, 6n − 4] ⎧ i ∈ [0, 2n − 1] ⎪ ⎪ 14i , ⎨ 14(n + 2i ) − 10 i ∈ [2n, 4n − 2], i even ⎪ 14(n + i −21 ) i ∈ [2n, 4n − 2], i odd ⎪ ⎩ 14(i − n) + 4, i ∈ [4n − 1, 6n − 3]  1, i ∈ [0, 4n − 3] 2, i ∈ [4n − 2, 5n − 3] 3, i ∈ [5n − 2, 6n − 4]† ⎧ ⎨ 3, i ∈ [0, n − 2]† 2, i ∈ [n − 1, 2n − 2] ⎩ 1, i ∈ [2n − 1, 6n − 4] ⎧ 1 − 2i , i ∈ [0, n − 1] ⎪ ⎪ ⎪ i ∈ [n, 2n − 1] ⎨ 2 − n − i, 3 − 3n, i ∈ [2n, 4n − 4]† ⎪ ⎪ i ∈ [4n − 3, 5n − 3] ⎪ 6 − 7n + i , ⎩ 9 − 12n + 2i , i ∈ [5n − 2, 6n − 4]† ⎧ −2i , i ∈ [0, n − 1] ⎪ ⎪ ⎪ i ∈ [n, 2n − 1]† ⎨ 1 − n − i, 2 − 3n, i ∈ [2n, 4n − 3]† ⎪ ⎪ 4 − 7n + i , i ∈ [4n − 2, 5n − 2] ⎪ ⎩ 6 − 12n + 2i , i ∈ [5n − 1, 6n − 3]

Manifold

Mi

mi

τ ( M i ) − τ (mi )

τ ( M i ) − τ (mi +1 )

τ (M i )

τ (mi )

  j

j = 1 + j −  =k+2−

  j

+

2

k+1 2

Clearly 0 = 1. Notice that



7

+

 +



j (5n + 1)

7







3, 2, 1,

even odd

even odd

i ∈ [0, n − 1] i ∈ [n, 2n − 1] i ∈ [2n, 6n + 1]

⎧ 1 − 2i , ⎪ ⎪ ⎪ ⎨ 1 − n − i, 1 − 3n, ⎪ ⎪ ⎪ −7n + i , ⎩ −1 − 12n + 2i , ⎧ −2i , ⎪ ⎪ ⎪ ⎨ −n − i , −3n, ⎪ ⎪ ⎪ ⎩ −2 − 7n + i , −4 − 12n + 2i ,

i ∈ [0, n] i ∈ [n + 1, 2n − 1] i ∈ [2n, 4n + 1] i ∈ [4n + 2, 5n + 1] i ∈ [5n + 2, 6n + 1] i ∈ [0, n] i ∈ [n + 1, 2n − 1]† i ∈ [2n, 4n + 2] i ∈ [4n + 3, 5n + 2] i ∈ [5n + 3, 6n + 2]



14n + 3

k+1

Σ(2, 7, 14n + 5) ⎧ 14i + 1, i ∈ [0, 2n] ⎪ ⎪ ⎨ 14(n + i ) + 1 i ∈ [2n + 1, 4n + 1], i 2 i −1 ⎪ ⎪ 14(n + 2 ) + 11 i ∈ [2n + 1, 4n + 1], i ⎩ i ∈ [4n + 2, 6n + 1] 14(i − n) + 1, ⎧ i ∈ [0, 2n] ⎪ ⎪ 14i , ⎨ 14(n + 2i ) i ∈ [2n + 1, 4n + 2], i ⎪ 14(n + i −21 ) + 10 i ∈ [2n + 1, 4n + 2], i ⎪ ⎩ 14(i − n) − 8, i ∈ [4n + 3, 6n + 2]  1, i ∈ [0, 4n + 1] 2, i ∈ [4n + 2, 5n + 1] 3, i ∈ [5n + 2, 6n + 1]

(k + 1)(5n + 1) − i + 14n + 3

 .

(12)

E. Tweedy / Topology and its Applications 160 (2013) 620–632

631

Fig. 2. Plumbing graphs for the Brieskorn homology spheres Σ(2, q, 2qn ± 1); unlabeled vertices have weight −2. For Σ(2, q, 2qn − 1), the graph shown is valid for n > 1; when n = 1, the rightmost arm only has the first 2q − 2 vertices.

Fig. 3. Plumbing graphs for the Brieskorn homology spheres Σ(2, 5, k) and Σ(2, 7, k). For Σ(2, 5, 10n − 3), Σ(2, 7, 14n − 3), and Σ(2, 7, 14n − 5), the graph shown is valid for n > 1 only; when n = 1, both the “(n − 2)-tail” and the next vertex inward are missing from the rightmost arm.

 

k+1 2 k+1

13 k =0 13

7

k =0

= (1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7), = (1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2).

Now in this case

(k + 1)(5n + 1) + 1 − 2n  (k + 1)(5n + 1) − i  (k + 1)(5n + 1), and one can directly show that



(k + 1)(5n + 1) − i 14n + 3

13 k =0

 = (1, 1, s, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5),

where s :=

2,

i ∈ [0, n − 1],

1,

i ∈ [n, 2n − 1].

In light of this, Eq. (12) gives that M

−1

( j ) j =i+M1i

 = (−1, 0, t , 0, 0, 0, 0, 0, −1, 0, 0, 0, 0, 1) where t :=

−1, i ∈ [0, n − 1], 0,

i ∈ [n, 2n − 1].

Case 2. Let M i  j < M i +1 , where 4n  i  6n + 1. Now we write j = 14(i − n) + 1 + k, where 0  k  13 and

(k + 1)(5n + 1) − 5n  (k + 1)(5n + 1) − (i − n)  (k + 1)(5n + 1) − 3n. Via an analysis similar to that in the previous case, one obtains that

(13)

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E. Tweedy / Topology and its Applications 160 (2013) 620–632

M −1 ( j ) j =i+M1i

 = (−1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, t , 0, 1) where t :=

0,

i ∈ [4n, 5n − 1],

1,

i ∈ [5n, 6n].

(14)

Case 3. Let M i  j < M i +1 , where 2n + 1  i  4n − 1 and i is odd. Now M i +1 − M i = 8, so we write j = 14(n + i −21 ) + 7 + k with 0  k  7. One then finds that M

−1

( j ) j =i+M1i



= (−1, 0, t , 0, 0, 0, 0, 0, −1, 0, 0, 0, 0, 1) where t :=

−1, i ∈ [0, n − 1], 0,

i ∈ [n, 2n − 1].

(15)

Case 4. Let M i  j < M i +1 , where 2n + 1  i  4n − 1 and i is even. Now M i +1 − M i = 6, so we write j = 14(n + 2i ) + 1 + k with 0  k  5. One then finds that M

−1

( j ) j =i+M1i

= (−1, 0, 0, 0, 0, 1).

(16)

Eqs. (13)–(16) imply that τ is indeed (non-strictly) decreasing on [ M i , mi +1 ] and (non-strictly) increasing on [mi , M i ] for all i ∈ [0, 6n]. It’s not hard to show that if j  m6n+1 that  j  0. 2 Proof of (ii). Keeping in mind the expressions for the M i and mi , one can use Eqs. (13)–(16) to obtain τ ( M i ) − τ (mi+1 ); these in turn give the expressions for τ ( M i ) and τ (mi ). 2

τ ( M i ) − τ (mi ) and

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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