Federico Brunero
email: brunero@eurecom.fr

Petros Elia
email: elia@eurecom.fr

The Exact Load-Memory Tradeoff of Multi-Access Coded Caching With Combinatorial Topology

Keywords: Coded caching, combinatorial topology, index coding, information-theoretic converse, multi-access coded caching (MACC)

Recently, Muralidhar et al. proposed a novel multiaccess system model where each user is connected to multiple caches in a manner that follows the well-known combinatorial topology of combination networks. For such multi-access topology, the same authors proposed an achievable scheme, which stands out for the unprecedented coding gains even with very modest cache resources. In this paper, we identify the fundamental limits of such multi-access setting with exceptional potential, providing an information-theoretic converse which establishes, together with the inner bound by Muralidhar et al., the exact optimal performance under uncoded prefetching.

I. INTRODUCTION

Coded caching was introduced in [START_REF] Maddah-Ali | Fundamental limits of caching[END_REF] as a coding-based communication technique with the purpose of reducing significantly the amount of data to be transferred from a centralized server to its cache-aided receiving users. The key idea behind coded caching involves an accurate joint design of the placement phase and the delivery phase. During the placement phase, which happens during off-peak hours, the caches of the users are preemptively filled without knowing the future requests. The caching is carefully performed so that, once the requests of the users are revealed, the amount of bits to be transferred during the delivery phase, which takes place when the network is saturated, is minimized. In the standard single-stream broadcast channel model with K receiving users, each of which is able to store in its cache a fraction γ of the main library, coded caching is able to provide a sizeable coding gain equal to Kγ + 1, where such coding gain represents simply the number of users to which a coded message is useful at the same time.

Current research on coded caching spans several topics such as the impact of multiple antennas on caching [START_REF] Parrinello | Fundamental limits of coded caching with multiple antennas, shared caches and uncoded prefetching[END_REF]- [START_REF] Tölli | Multi-antenna interference management for coded caching[END_REF], the interplay between caching and file popularity [START_REF] Zhang | Coded caching under arbitrary popularity distributions[END_REF]- [START_REF] Hachem | Coded caching for multi-level popularity and access[END_REF], and a variety of other scenarios [START_REF] Joudeh | Fundamental limits of wireless caching under mixed cacheable and uncacheable traffic[END_REF]- [START_REF] Zhao | Vector coded caching multiplicatively boosts the throughput of realistic downlink systems[END_REF]. For a thorough review of the existing coded caching works, we strongly encourage the reader to refer to the longer version of this work [START_REF] Brunero | Fundamental limits of combinatorial multi-access caching[END_REF].

A. Multi-Access Coded Caching

Differently from the model in [START_REF] Maddah-Ali | Fundamental limits of caching[END_REF] where each user has access to its own single dedicated cache, it is conceivable that in This work was supported by the European Research Council (ERC) through the EU Horizon 2020 Research and Innovation Program under Grant 725929 (Project DUALITY). An extended version of this work [START_REF] Brunero | Fundamental limits of combinatorial multi-access caching[END_REF] has been submitted to IEEE Transactions on Information Theory. several scenarios each cache serves more than one user, and that each user can connect to more than one cache. For instance, in dense cellular networks, the cache-aided access points (APs) could have overlapping coverage areas, allowing in this way each user to connect to more than one AP. Such scenario motivated the work in [START_REF] Hachem | Coded caching for multi-level popularity and access[END_REF], where the authors introduced a new parameter λ to keep into consideration the number of caches that each user can access. The corresponding model defined by λ as well as by the number of users K, the number of library files N , and the cache size of M files was referred to as the multi-access coded caching (MACC) model. Such model includes Λ caches and K = Λ users, where each of them is connected to λ > 1 consecutive caches in a cyclic wrap-around fashion.

Since the introduction of the aforementioned multi-access model with cyclic wrap-around topology, various works focused on the design of coding schemes that leverage the multi-access nature of the problem. For instance, a caching-and-delivery scheme was first proposed in the original work [START_REF] Hachem | Coded caching for multi-level popularity and access[END_REF] with a decentralized (stochastic) cache placement, where this scheme provided improved local caching gains. Another achievable scheme preserving both the full local caching gain and the optimal coding gain of Λλγ + 1 was instead presented in [START_REF] Serbetci | Multi-access coded caching: gains beyond cache-redundancy[END_REF], albeit for the rather unrealistically demanding scenario where λ = (Λ -1)/Λγ. Another notable work is then [START_REF] Reddy | Rate-memory trade-off for multiaccess coded caching with uncoded placement[END_REF], where the authors designed a novel scheme for any λ ≥ 1, which was proved, for the similarly demanding regime of λ ≥ Λ/2 and Λγ ≤ 2, to be at a factor of at most 2 from the optimal under the assumption of uncoded placement. Other relevant works investigated the MACC problem and its connection to topics such as privacy and secrecy [START_REF] Namboodiri | Multi-access coded caching with secure delivery[END_REF], [START_REF]Multi-access coded caching with demand privacy[END_REF], structured index coding problems [START_REF] Reddy | Structured index coding problem and multi-access coded caching[END_REF] and PDA designs [START_REF] Cheng | A novel transformation approach of shared-link coded caching schemes for multiaccess networks[END_REF].

Recently, some interest arose towards new multi-access models which deviate from the cyclic wrap-around topology in [START_REF] Hachem | Coded caching for multi-level popularity and access[END_REF]. For example, a new MACC paradigm was presented in [START_REF] Katyal | Multi-access coded caching schemes from cross resolvable designs[END_REF], which involved topologies that are inspired by cross resolvable designs (CRDs), a special class of designs in combinatorics. However, a substantial breakthrough came with the work in [START_REF] Muralidhar | Maddah-Ali-Niesen scheme for multi-access coded caching[END_REF], where the authors proposed a MACC model enjoying the same amount of resources λ and Λγ, but where now the users and the caches are connected following the wellknown combinatorial topology of combination networks [START_REF] Ji | On the fundamental limits of caching in combination networks[END_REF]. This was a breakthrough because it allowed for the deployment of a subsequent scheme, presented in [START_REF] Muralidhar | Maddah-Ali-Niesen scheme for multi-access coded caching[END_REF] as a generalization of the original Maddah-Ali and Niesen (MAN) scheme in [START_REF] Maddah-Ali | Fundamental limits of caching[END_REF], that achieves the astounding coding gain Λγ+λ λ far exceeding Λγ + 1 even for small values of λ and Λγ, which is the regime that really matters.

B. Main Contributions

Our work explores the fundamental limits of this undoubtedly powerful MACC model by Muralidhar et al., which based on the findings in [START_REF] Muralidhar | Maddah-Ali-Niesen scheme for multi-access coded caching[END_REF] has astounding performance. Such fundamental limits had remained entirely unknown as no information-theoretic converse has ever been developed. We here establish the exact optimal performance with the introduction of a novel information-theoretic lower bound on the optimal worst-case communication load under the assumption of uncoded placement. Further results are presented in [START_REF] Brunero | Fundamental limits of combinatorial multi-access caching[END_REF], such as a generalization of the achievable scheme in [START_REF] Muralidhar | Maddah-Ali-Niesen scheme for multi-access coded caching[END_REF] as well as novel information-theoretic converses for the topologyagnostic multi-access setting, i.e., the setting where it is not known a priori how the K users are connected to the Λ caches in the system.

C. Paper Outline

The paper is organized as follows. The MACC model and some preliminary definitions are presented in Section II. Section III presents the information-theoretic converse, whereas its general proof is described in Section IV. Section V concludes the paper. The appendices hold supplementary material.

D. Notation

We denote by Z + the set of positive integers. For n ∈ Z + , we define [n] := {1, . . . , n}. If a, b ∈ Z + such that a < b, then [a : b] := {a, a + 1, . . . , b -1, b}. For sets we use calligraphic symbols, whereas for vectors we use bold symbols. Given a finite set A, we denote by |A| its cardinality. We denote by n k the binomial coefficient and we let n k = 0 whenever n < 0, k < 0 or n < k.

II. SYSTEM MODEL

We consider the centralized coded caching scenario where one single server has access to a library L = {W n : n ∈ [N ]} containing N files of B bits each. The server is connected to K users through an error-free broadcast link. In the system there are Λ caches, each of size M B bits. In agreement with the system model in [START_REF] Muralidhar | Maddah-Ali-Niesen scheme for multi-access coded caching[END_REF], each user is connected exactly and uniquely to a subset of λ caches for some fixed value of λ ∈ [Λ], which consequently implies that there are K = Λ λ users for any given Λ and λ ∈ [Λ]. We denote 1 by U ⊆ [Λ] the user connected to the |U| = λ caches in the set U. We further assume that the link between the server and the users is the main bottleneck, whereas we assume that the channel between each user and its assigned caches has infinite capacity. As is common, we assume that N ≥ K. Such setting is completely described by the tuple (Λ, λ, N, M ) and we refer to it as the MACC problem with combinatorial topology. 1 For ease of notation, we will often omit braces and commas when indicating sets. For instance, user {1, 2} in Fig. 1 is denoted as 12. The communication procedure works as follows. During the placement phase, which typically occurs well before the delivery phase, the central server fills the caches without any knowledge of the future requests from the users. The delivery phase, which typically happens when the network is saturated and interference-limited, commences when the file-requests of all users are simultaneously revealed. During delivery, the server prepares some coded messages which are sent over the bottleneck shared link, so that each user can retrieve the missing information from the received transmission. The users cancel the interference terms that appear in the broadcast transmission, and do so by means of the cached contents they have access to, eventually decoding their own messages. As a consequence, the worst-case communication load R is defined as the total number of transmitted bits, normalized by the file-size B, that can guarantee the correct delivery of any K-tuple of requested files in the worst-case scenario. The optimal communication load R ⋆ is formally defined as

R ⋆ (M ) := inf{R : (M, R) is achievable} (1)
where the tuple (M, R) is said to be achievable if there exists a caching-and-delivery procedure for which, for any possible demand, a load R can be guaranteed for a given memory value M . We use the notation W d U to denote the file requested by the user identified by U for some U ⊆ [Λ] with |U| = λ, which we remind the reader is simply the user connected exactly and uniquely to the λ caches in the set U. Our goal is to provide a converse bound on the optimal worst-case load under the assumption of uncoded placement, whose definition is given in the following.

Definition 1 (Uncoded Cache Placement). A cache placement is uncoded if the bits of the files are simply copied within the caches of the users.

Denoting by R ⋆ u the optimal worst-case load under uncoded placement, it trivially holds R ⋆ u ≥ R ⋆ .

III. MAIN RESULT

We present the main contribution of the paper. The derivation of the converse bound adopts the index coding technique first proposed in [START_REF] Wan | An index coding approach to caching with uncoded cache placement[END_REF]. Our main challenge will be to design the converse in such a way that it tightly captures the fact that each user, differently from the original setting in [START_REF] Maddah-Ali | Fundamental limits of caching[END_REF], is connected to λ caches for some λ ∈ [Λ]. The result is stated in the following theorem.

Theorem 1. Consider the multi-access coded caching problem with combinatorial topology with parameters (Λ, λ, N, M ). Under the assumption of uncoded cache placement, the optimal worst-case communication load R ⋆ u is a piecewise linear curve with corner points

(M, R ⋆ u ) = t N Λ , Λ t+λ Λ t , ∀t ∈ [0 : Λ]. (2)
Proof. The proof of achievability comes from the coding scheme in [START_REF] Muralidhar | Maddah-Ali-Niesen scheme for multi-access coded caching[END_REF], whereas the proof of the converse bound is presented in Section IV.

The longer version in [START_REF] Brunero | Fundamental limits of combinatorial multi-access caching[END_REF] of this work provides also another interesting result that we will simply mention here. The system model in [START_REF] Muralidhar | Maddah-Ali-Niesen scheme for multi-access coded caching[END_REF] considers a single value of λ ∈ [Λ] for a given Λ, although such setting can be extended into a generalized combinatorial multi-access model which supports the coexistence of groups of users connected to different numbers of caches. This extension can be obtained if we consider a simultaneous coexistence of each model in [START_REF] Muralidhar | Maddah-Ali-Niesen scheme for multi-access coded caching[END_REF] for each λ ∈ [Λ]. The quite surprising outcome in [START_REF] Brunero | Fundamental limits of combinatorial multi-access caching[END_REF] is summarized in the following remark. Remark 1. For the MACC problem with generalized combinatorial topology, there is no need to encode over users which are connected to different numbers of caches, even though there is abundance of coding opportunities. Instead, applying the coding scheme in [START_REF] Muralidhar | Maddah-Ali-Niesen scheme for multi-access coded caching[END_REF] in a TDMA-like manner is sufficient to achieve the optimal communication load under uncoded prefetching. An additional powerful insight that comes out of this is that the basic Λ-cache MAN placement proves to be extremely effective as it allows for the optimal performance for any instance of the generalized combinatorial topology.

We now proceed by presenting the converse proof of Theorem 1.

IV. PROOF OF THE CONVERSE BOUND

The converse relies on the well-known acyclic subgraph index coding bound, which has been extensively used in various other settings (see for example [START_REF] Parrinello | Fundamental limits of coded caching with multiple antennas, shared caches and uncoded prefetching[END_REF], [START_REF] Wan | An index coding approach to caching with uncoded cache placement[END_REF], [START_REF] Brunero | Unselfish coded caching can yield unbounded gains over symmetrically selfish caching[END_REF] to name a few) in order to derive lower bounds on the optimal worst-case load in caching under the assumption of uncoded prefetching. To clarify the connection between this bound and our setting, we start by providing a brief presentation of the index coding problem and its connection to coded caching.

A. Definition of the Index Coding Problem

The index coding problem [START_REF] Bar-Yossef | Index coding with side information[END_REF] consists of a central server having access to N ′ independent messages, and of K ′ users that are connected to the server via a shared error-free broadcast channel. Each user k ∈ [K ′ ] has a set of desired messages M k ⊆ [N ′ ], which is called the desired message set, while also having access to another subset of messages A k ⊆ [N ′ ], which is called the side information set. To avoid trivial scenarios, it is commonly assumed that

M k ̸ = ∅, A k ̸ = [N ′ ] and M k ∩A k = ∅ for each k ∈ [K ′ ].
The index coding problem is usually described in terms of its side information graph. Let M i be the i-th message for some i ∈ [N ′ ]. Then, such graph is a directed graph where each vertex is a desired message and where there exists an edge from a desired message M i to a desired message M j if and only if message M i is in the side information set of the user requesting message M j . The derivation of our converse is based on the following bound from [START_REF] Arbabjolfaei | On the capacity region for index coding[END_REF]Corollary 1].

Lemma 1 ([24, Corollary 1]). Consider an index coding problem with N

′ messages M i for i ∈ [N ′ ].
The minimum number of transmitted bits ρ is lower bounded as

ρ ≥ i∈J |M i | (3)
for any acyclic subgraph J of the side information graph.

B. Main Proof

The first step in our converse proof consists of dividing, in the most generic manner, each file into a maximum of 2 Λ disjoint subfiles as

W n = {W n,T : T ⊆ [Λ]} , ∀n ∈ [N ] (4)
where we identify with W n,T the subfile which is exclusively stored by the caches in T . Noticing that the bits of the library files are simply copied within the caches, such placement is uncoded according to Definition 1. 1) Constructing the Index Coding Bound: Assuming that each user requests a distinct 2 file, we consider the index coding problem with K ′ = K = Λ λ users and N ′ = Λ λ 2 Λ-λ independent messages, where each such message represents a subfile requested by some user (who naturally does not have access to it via a cache). Recalling that W d U denotes the file requested by the user identified by U, the desired message set and the side information set are respectively given, in their most generic form, by

M U = {W d U ,T : T ⊆ ([Λ] \ U)} (5) A U = {W n,T : n ∈ [N ], T ⊆ [Λ], T ∩ U ̸ = ∅} (6)
for each user U with U ⊆ [Λ] and |U| = λ. Here, the side information graph consists of a directed graph where each vertex is a subfile, and where there is an edge from the subfile

W d U 1 ,T1 to the subfile W d U 2 ,T2 if and only if W d U 1 ,T1 ∈ A U2 with U j ⊆ [Λ], |U j | = λ, T j ⊆ ([Λ] \ U j ) for j ∈ {1, 2} and U 1 ̸ = U 2 .
Since our aim is to apply Lemma 1, we need to consider acyclic sets of vertices J in the side information graph. Toward this, we take advantage of the following lemma.

W d U i ,Ti (7)
is acyclic.

Proof. The proof is provided in Appendix A. An illustrative example is then described in Appendix B.

Consider a demand vector d and a permutation c of the set [Λ]. Applying Lemma 2 yields the following lower bound

BR ⋆ u ≥ R(d, c) (8)
where R(d, c) is defined as

R(d, c) := Λ i=λ U i ⊆{c1,...,ci}:|U i |=λ, ci∈U i Ti⊆{ci+1,...,c K } W d U i ,Ti . (9) 2
) Constructing the Optimization Problem: Now our goal is to create several bounds as the one in (8) considering any vector d ∈ D and any vector c ∈ C, where we denote by D and C the set of possible demand vectors with distinct entries and the set of possible permutation vectors of the set [Λ], respectively. Our aim is then to average all these bounds to obtain in the end a useful lower bound on the optimal worstcase load. Considering that |D| = N K K! and |C| = Λ!, we aim to simplify the expression given by

N K K!Λ!BR ⋆ u ≥ d∈D c∈C R(d, c). (10)
| = λ!t ′ !(Λ-λ-t ′ )! Λ t ′ +λ . Hence, the subfile W n,T is counted a total of λ!t ′ !(Λ -λ -t ′ )! Λ t ′ +λ N K K!/
a t ′ = Λ -t ′ λ λ!t ′ !(Λ -λ -t ′ )! Λ t ′ + λ N K K! N (11)
times, which gives us the number of times this same subfile appears in [START_REF] Zhao | Vector coded caching multiplicatively boosts the throughput of realistic downlink systems[END_REF]. The same reasoning follows for any n ∈ [N ] and for any T ⊆ [Λ] with |T | = t ′ . Thus, the expression in [START_REF] Zhao | Vector coded caching multiplicatively boosts the throughput of realistic downlink systems[END_REF] can be rewritten as

R ⋆ u ≥ 1 N K K!Λ! Λ t ′ =0 N a t ′ x t ′ (12) = Λ t ′ =0 Λ t ′ +λ Λ t ′ x t ′ (13)
= Λ t ′ =0 f (t ′ )x t ′ (14)
where f (t ′ ) and x t ′ are defined as

f (t ′ ) := Λ t ′ +λ Λ t ′ (15) 0 ≤ x t ′ := n∈[N ] T ⊆[Λ]:|T |=t ′ |W n,T | N B . (16)
At this point, we seek to lower bound the minimum worstcase load R ⋆ u by lower bounding the solution to the following optimization problem

min x Λ t ′ =0 f (t ′ )x t ′ (17a) subject to Λ t ′ =0 x t ′ = 1 (17b) Λ t ′ =0 t ′ x t ′ ≤ ΛM N (17c)
where (17b) and (17c) correspond to the file-size constraint and the cumulative cache-size constraint, respectively.

3 Indeed, the subfile W n,T appears in the acyclic graph chosen as in Lemma 2 for all those permutations c = (c 1 , . . . , c Λ ) for which U = U i and T = T i for some i ∈ [λ : Λ] such that U i ⊆ {c 1 , . . . , c i } with |U i | = λ and c i ∈ U i , and such that T i ⊆ {c i+1 , . . . , c Λ }, i.e., this happens whenever the elements in T are after the elements in U in the permutation vector c.

3) Lower Bounding the Solution to the Optimization Problem: Since the auxiliary variable x t ′ can be considered as a probability mass function, the optimization problem in [START_REF] Cheng | A novel transformation approach of shared-link coded caching schemes for multiaccess networks[END_REF] can be seen as the minimization of E[f (t ′ )]. Moreover, the following holds. Lemma 3. The function f (t ′ ) is convex and decreasing in t ′ .

Proof. The proof is relegated to the longer version of this work [START_REF] Brunero | Fundamental limits of combinatorial multi-access caching[END_REF], where the proof is provided for a generalization of the function here denoted by f (t ′ ).

Taking advantage of Lemma 3, we can write E[f (t ′ )] ≥ f (E[t ′ ]) using Jensen's inequality. Then, considering that f (t ′ ) is also decreasing with increasing t ′ ∈ [0 : Λ], we can further write f (E[t ′ ]) ≥ f (ΛM/N ) taking advantage of the fact that E[t ′ ] is upper bounded as in (17c). Consequently, E[f (t ′ )] ≥ f (ΛM/N ), and thus for t := ΛM/N the optimal worst-case load R ⋆ u is lower bounded by R LB which is a piecewise linear curve with corner points

(M, R LB ) = t N Λ , Λ t+λ Λ t , ∀t ∈ [0 : Λ]. (18)
This concludes the proof.

V. CONCLUSION

In this work, we derived the fundamental limits of a coded caching scenario with exceptional potential. We proposed a novel information-theoretic converse that manages to capture the topological properties of the multi-access model in Section II. The lower bound matches the achievable performance of the coding scheme in [START_REF] Muralidhar | Maddah-Ali-Niesen scheme for multi-access coded caching[END_REF], so allowing us to identify the exact optimal performance of multi-access caching with combinatorial topology under uncoded prefetching. Interestingly, our information-theoretic converse can be seen as a generalization of the lower bound in [START_REF] Wan | An index coding approach to caching with uncoded cache placement[END_REF], which similarly proved the exact optimality of the MAN placement-and-delivery scheme.

As already mentioned, the longer version of this work in [START_REF] Brunero | Fundamental limits of combinatorial multi-access caching[END_REF] provides a variety of interesting extensions. Indeed, the work in [START_REF] Brunero | Fundamental limits of combinatorial multi-access caching[END_REF] not only provides a generalization of the combinatorial topology that allows for the coexistence of users connected to different numbers of caches, but also provides very interesting results regarding the topology-agnostic multiaccess problem, offering novel lower bounds on the average worst-case performance when it is not known a priori how the K users are connected to the Λ caches in the system.

W d U i ,Ti (19)
in Lemma 2. We can show that such set is guaranteed to be acyclic by following the same reasoning as in the proof of [21, Lemma 1].

For a specific permutation of caches c = (c 1 , . . . , c Λ ), we will say that the subfile W d U i ,Ti belongs to the i-th level4 , which will mean that U i ⊆ {c 1 , . . . , c i } with |U i | = λ and c i ∈ U i , and that T i ⊆ ([Λ] \ {c 1 , . . . , c i }) with i ∈ [λ : Λ]. As one may see, no user in the i-th level has access to the subfiles in its level, since indeed U i ∩T i = ∅ for each U i ⊆ {c 1 , . . . , c i } with |U i | = λ and c i ∈ U i , and for each T i ⊆ ([Λ] \ {c 1 , . . . , c i }). Moreover, no user in the i-th level has access to the subfiles in higher levels, since U i ∩T j = ∅ for each U i ⊆ {c 1 , . . . , c i } with |U i | = λ and c i ∈ U i , and for each T j ⊆ ([Λ] \ {c 1 , . . . , c j }) with j ∈ [i + 1 : Λ]. Consequently, we can conclude that the set in [START_REF] Muralidhar | Maddah-Ali-Niesen scheme for multi-access coded caching[END_REF] is acyclic. This concludes the proof.

Fig. 1 .

 1 Fig. 1. MACC problem with combinatorial topology and Λ = 4 caches, where each user is connected exactly and uniquely to a subset of λ = 2 caches.

  For the sake of simplicity, we denote by d = (d U : U ⊆ [Λ], |U| = λ) the demand vector containing the indices of the files requested by the users in the system, i.e., d U ∈ [N ] for each U ⊆ [Λ] with |U| = λ. The following example can help familiarize the reader with the setting. Example 1 (Λ = 4, λ = 2, N, M ). Consider the MACC problem with combinatorial topology and Λ = 4 caches in Fig. 1. Since each set of λ = 2 caches is uniquely assigned to a user, there are K = Λ λ = 6 users in total.Recalling that each user is identified by the set of 2 caches it is connected to and that for simplicity we omit braces and commas when indicating sets, we let 12 represent the user connected to cache 1 and cache 2, we let 13 represent the user connected to cache 1 and cache 3, and so on. The demand vector is given by d = (d 12 , d 13 , d 14 , d 23 , d 24 , d 34 ), where d 12 ∈ [N ] is the index of the file requested by the user 12, d 13 ∈ [N ] is the index of the file requested by the user 13, and so on.

Lemma 2 .

 2 Let d = (d U : U ⊆ [Λ], |U| = λ) be a demand vector and let c = (c 1 , . . . , c Λ ) be a permutation of the Λ caches. The following set of vertices i∈[λ:Λ] U i ⊆{c1,...,ci}:|U i |=λ, ci∈U i Ti⊆([Λ]\{c1,...,ci})

Toward simplifying ( 10 )

 10 , we proceed by counting how many times each subfile W n,T -for any given n ∈ [N ], T ⊆ [Λ] and |T | = t ′ for some t ′ ∈ [0 : Λ] -appears in[START_REF] Zhao | Vector coded caching multiplicatively boosts the throughput of realistic downlink systems[END_REF].Let us focus on the subfile W n,T for some n ∈ [N ], T ⊆ [Λ] and |T | = t ′ with t ′ ∈ [0 : Λ]. Assume that the file W n is demanded by user U for some U ⊆ ([Λ] \ T ) with |U| = λ and denote by D n,U the set of demands such that d U = n. Out of the entire set D of all possible distinct demands, we find a total of N K K!/N distinct demands for which a file is requested by the same user. Hence, it holds |D n,U | = N K K!/N . For each d ∈ D n,U and for each c ∈ C there is a corresponding boundR(d, c). The subfile W n,T appears only in the bounds induced by permutation vectors c ∈ C such that the elements in the set U appear in the vector c before 3 the elements in the set T . If we denote by C U ,T the set of such permutation vectors, it can be verified that |C U ,T

  N times when considering the bounds R(d, c) with d ∈ D n,U and c ∈ C U ,T . Since the same reasoning follows for each U ⊆ ([Λ] \ T ) with |U| = λ, the subfile W n,T is counted a total of

  of vertices i∈[λ:Λ] U i ⊆{c1,...,ci}:|U i |=λ, ci∈U i Ti⊆([Λ]\{c1,...,ci})

The set of worst-case demands may not include the set of demand vectors d with all distinct entries. However, since our goal is to derive a converse bound on the worst-case load, this is not a problem. Indeed, the choice of treating distinct demands yields a valid converse bound, since such bound does not need to be, a priori, the tightest bound. In our case, it proves to be tight.

The term level carries the same meaning as in the proof of [21, Lemma 1], and its impact here is described mathematically in compliance with our setting.

ILLUSTRATIVE EXAMPLE

To illustrate the main idea in Lemma 2, we provide in the following a simple example where we explicitly construct the acyclic set of vertices for two different permutations of caches.

Consider the setting where there are Λ = 4 caches and each user is connected to λ = 2 caches, which implies K = 6 users under the combinatorial topology. Assume the arbitrary demand vector d = (d 12 , d 13 , d 14 , d 23 , d 24 , d 34 ) and consider the permutation of caches c = [START_REF] Maddah-Ali | Fundamental limits of caching[END_REF][START_REF] Tölli | Multi-antenna interference management for coded caching[END_REF][START_REF] Lampiris | Resolving the feedback bottleneck of multi-antenna coded caching[END_REF][START_REF] Parrinello | Fundamental limits of coded caching with multiple antennas, shared caches and uncoded prefetching[END_REF]. According to Lemma 2, when i = 2, for the user identified by the set {1, 4} we pick all the subfiles W d14,T2 where T 2 ⊆ {2, 3}, i.e., we pick {W d14,∅ , W d14,2 , W d14,3 , W d14,23 }. When i = 3, for each user U 3 ∈ {{1, 3}, {3, 4}} we pick all the subfiles W d U 3 ,T3 where T 3 ⊆ {2}, i.e., we pick {W d13,∅ , W d13,2 , W d34,∅ , W d34,2 , }. When i = 4, for each user U 4 ∈ {{1, 2}, {2, 4}, {2, 3}} we pick all the subfiles W d U 4 ,T4 where T 4 = ∅, i.e., we pick {W d12,∅ , W d24,∅ , W d23,∅ }. Hence, for the permutation vector c = (1, 4, 3, 2) the acyclic set of vertices from Lemma 2 is given by the set

Consider now the cache permutation c = (2, 3, 1, 4). When i = 2, for the user identified by the set {2, 3} we pick all the subfiles W d23,T2 where T 2 ⊆ {1, 4}, i.e., we pick {W d23,∅ , W d23,1 , W d23,4 , W d23,14 }. When i = 3, for each user U 3 ∈ {{1, 2}, {1, 3}} we pick all the subfiles W d U 3 ,T3 where T 3 ⊆ {4}, i.e., we pick {W d12,∅ , W d12,4 , W d13,∅ , W d13,4 }. When i = 4, for each user U 4 ∈ {{2, 4}, {3, 4}, {1, 4}}, we pick all the subfiles W d U 4 ,T4 where T 4 = ∅, i.e., we pick {W d24,∅ , W d34,∅ , W d14,∅ }. Hence, for the permutation vector c = (2, 3, 1, 4) the acyclic set of vertices from Lemma 2 is given by the set {W d23,∅ , W d23,1 , W d23,4 , W d23,14 , W d12,∅ , W d12,4 , W d13,∅ , W d13,4 , W d24,∅ , W d34,∅ , W d14,∅ }. [START_REF] Wan | An index coding approach to caching with uncoded cache placement[END_REF]


Vincent Blanloeil
email: v.blanloeil@math.unistra.fr

COBORDANT ALGEBRAIC KNOTS DEFINED BY BRIESKORN POLYNOMIALS

Keywords: Mathematics Subject Classification. Primary 57Q45; Secondary 57Q60, 32S55 Knot cobordism, algebraic knot, Brieskorn singularity, Seifert form, Witt equivalence

. We define new sets of invariants for these cobordism classes. Using these invariants we find more examples of distinct cobordism classes with distinct exponents.

Introduction

A Brieskorn polynomial is a polynomial of the form

P (z) = z a1 1 + z a2 2 + • • • + z an+1 n+1
with z = (z 1 , z 2 , . . . , z n+1 ), n ≥ 1, where the integers a j ≥ 2, j = 1, 2, . . . , n + 1, are called the exponents. The complex hypersurface in C n+1 defined by P = 0 has an isolated singularity at the origin, which is called a Brieskorn singularity.

To be more precise, let f : (C n+1 , 0) → (C, 0) be a holomorphic function germ with an isolated critical point at the origin. We denote by D 2n+2 ε the closed ball of radius ε > 0 centred at 0 in C n+1 , and by S 2n+1 ε its boundary. According to Milnor [START_REF] Milnor | Singular points of complex hypersurfaces[END_REF], the oriented homeomorphism class of the pair (D 2n+2 ε , f -1 (0) ∩ D 2n+2 ε ) does not depend on the choice of a sufficiently small ε > 0, and by definition it is the topological type of f . The oriented diffeomorphism class of the pair (S 2n+1 ε , K f ), with K f = f -1 (0) ∩ S 2n+1 ε , is the algebraic knot associated with f , where K f is a closed oriented (2n-1)-dimensional manifold. According to Milnor's cone structure theorem [START_REF] Milnor | Singular points of complex hypersurfaces[END_REF], the algebraic knot K f determines the topological type of f . In fact, it is known that the converse also holds. Definition 1.1. An m-dimensional knot, or a m-knot, is a closed oriented mdimensional submanifold of the oriented (m + 2)-dimensional sphere S m+2 . When this submanifold is homeomorphic to a sphere we call this a sperical knot. Two m-knots K 0 and K 1 in S m+2 are said to be cobordant if there exists a properly embedded oriented (m + 1)-dimensional submanifold X of S m+2 × [0, 1] such that (1) X is diffeomorphic to K 0 × [0, 1], and

(2) ∂X = (K 0 × {0}) ∪ (-K 1 × {1}), where -K 1 × {1} denotes the manifold K 1 × {1} with the reversed orientation. A manifold X as above is called a cobordism between K 0 and K 1 (see Fig. 1).

In this paper, we will study the cobordism classes of algebraic knots associated with Brieskorn singularities, we will call such knots Brieskorn knots for short.

r r K0 S m+2 × {0} r r K1 S m+2 × {1} S m+2 × [0, 1] Figure 1. A cobordism between K 0 and K 1
Moreover, our goal is to study these cobordism classes in terms of exponents of the Brieskorn polynomials. In [START_REF] Blanloeil | Cobordism of algebraic knots defined by Brieskorn polynomials[END_REF] we proved that when n equals 1 or 2 Brieskorn knots are cobordant if and only if they have same exponents. Then, we will work with n ≥ 3.

In [START_REF] Blanloeil | A theory of cobordism for non-spherical links[END_REF], for n ≥ 3, necessary and sufficient conditions for two algebraic (2n -1)knots to be cobordant have been obtained in terms of Seifert forms (for the definition of the Seifert form, see §2). However, the computation of the Seifert form of a given algebraic knot is very difficult, and an explicit calculation of a Seifert form is known only for a very limited class of algebraic knots. Furthermore, even if we know the Seifert forms explicitly, it is still difficult to see if given two such forms satisfy the algebraic conditions given in [START_REF] Blanloeil | A theory of cobordism for non-spherical links[END_REF] or not.

When the knots are spherical, the algebraic condition of cobordism becomes much more simple. M. Kervaire [START_REF] Kervaire | Knot cobordism in codimension two[END_REF] and J. Levine [START_REF] Levine | Knot cobordism groups in codimension two[END_REF] proved that spherical knot are cobordant if and only if they have Witt equivalent Seifert forms and J. Levine [START_REF] Levine | Invariants of knot cobordism[END_REF] gave a complete list of invariants for cobordism classes of spherical knots. (for details, see §2).

Recall that cobordism does not necessarily imply isotopy for algebraic knots in general. For details, see the survey article [START_REF] Blanloeil | Cobordism of fibered knots and related topics[END_REF].

In this paper, we first associate some spherical Brieskorn knots to a given Brieskorn knot. Since the set of exponents of such associated spherical knots are very similar to which of the Brieskorn knot, the study of cobordism classes of these spherical Brieskorn knots impose conditions on the exponents' sets of the initial Brieskorn cobordism class.

The paper is organized as follows. In §2 we give some definitions and classical results, then in §3 we give a new set of invariants for cobordism classes of Brieskorn knots.

Throughout the paper we work in the smooth category. All the homology groups are with integer coefficients unless otherwise specified.

Definitions and results on spherical knots cobordism

Let f (z) be a polynomial in C n+1 with an isolated critical point at the origin. We denote by F f the Milnor fiber associated with f , i.e., F f is the closure of a fiber of the Milnor fibration ϕ f :

S 2n+1 ε \ K f → S 1 defined by ϕ f (z) = f (z)/|f (z)|. According to Milnor [12], F f is a compact 2n-dimensional submanifold of S 2n+1 ε ✛ + ✂ ✂ ✍ ★ ✥ ✧ ✦ ★ ✥ ✧ ✦ ξ ✲ ξ + ✲ η ✲ η + ✲ F Figure 2.
Computing a Seifert matrix for the trefoil knot which is homotopy equivalent to the bouquet of a finite number of copies of the n-dimensional sphere. The Seifert form

L f : H n (F f ) × H n (F f ) → Z associated with f is defined by L f (α, β) = lk(a + , b),
where a and b are n-cycles representing α and β in H n (F f ) respectively, a + is the n-cycle in S 2n+1 ε obtained by pushing a into the positive normal direction of F f , and lk denotes the linking number of n-cycles in S 2n+1 ε . See Fig. 2 for a picture of cycles necessary to compute the trefoil knot's Seifert from. It is known that the isomorphism class of the Seifert form is a topological invariant of f . Furthermore, two algebraic knots K f and K g associated with polynomials f and g in C n+1 , respectively, with isolated critical points at the origin are isotopic in S 2n+1 ε if and only if their Seifert forms L f and L g are isomorphic, provided that n ≥ 3.

In fact, algebraic knots are simple fibered knots as follows. We say that an oriented m-knot K is fibered if there exists a smooth fibration φ : S m+2 \ K → S 1 and a trivialization τ :

N K → K × D 2 of a closed tubular neighborhood N K of K in S m+2 such that φ| NK \K coincides with π • τ | NK \K , where π : K × (D 2 \ {0}) → S 1
is the composition of the projection to the second factor and the obvious projection D 2 \ {0} → S 1 . Note that then the closure of each fiber of φ in S m+2 is a compact (m + 1)-dimensional oriented manifold whose boundary coincides with K. We shall often call the closure of each fiber simply a fiber. Moreover, for m = 2n -1 ≥ 1 we say that a fibered (2n -1)-knot K is simple if each fiber of φ is (n -1)-connected and K is (n -2)-connected. For details we refer the reader to [START_REF] Blanloeil | Cobordism of fibered knots and related topics[END_REF]. Note that two simple fibered (2n -1)-knots are isotopic if and only if they have isomorphic Seifert forms, provided n ≥ 3 (see [START_REF] Durfee | Fibered knots and algebraic singularities[END_REF][START_REF] Kato | A classification of simple spinnable structures on a 1-connected Alexander manifold[END_REF]).

Definition 2.1. Two bilinear forms L i : G i × G i → Z, i = 0, 1, defined on free abelian groups G i of finite ranks are said to be Witt equivalent if there exists a direct summand M of G 0 ⊕ G 1 such that (L 0 ⊕ (-L 1 ))(x, y) = 0 for all x, y ∈ M and twice the rank of M is equal to the rank of G 0 ⊕ G 1 . In this case, M is called a metabolizer.

2.1.

Characterization of spherical Brieskorn knots. We refer to [START_REF] Brieskorn | Beispiele zue Diffenrentialtopologie von Singularitaten[END_REF] for the results of this subsection, the reader may consult the book [START_REF] Dimca | Singularities and topology of hypersurfaces[END_REF] for detailed proofs as well.

Let P (z) = z a1 1 + z a2 2 + • • • + z an+1 n+1
be a Brieskorn polynomial, we associate to this polynomial a graph G P . This graph has n + 1 vertices labeled by the letters a 1 , . . . , a n+1 , and two vertices a i and a j are connected with and edge if and only if gcd(a i , a j ) is strictly greater to 1. We denote by C ev,P the connected component of G P wich contain all odd exponents ; remark that C ev,P may contain some odd labeled vertices as well. We say that C ev,P fulfills conditionC if it contains an odd number of vertices and gcd(a i , a j ) = 2 for any two distinct points a i and a j in C ev,P .

Theorem 2.2 ([4]

). Let n ≥ 3. The Brieskorn knot associated to P is spherical if and only if the graph G P contains at least two isolated points or an isolated point which is odd and the component C ev,P fulfills conditionC.

Let K P be the Brieskorn knot associated to the Brieskorn polynomial

P (z) = z a1 1 + z a2 2 + • • • + z an+1 n+1
. Set p and q be two distinct prime numbers such that for all i = 1, . . . , n + 1 we have gcd(p, a i ) = gcd(q, a i ) = gcd(p, q) = 1. Then we define the following polynomial

P p,q (z) = z a1 1 + z a2 2 + • • • + z an+1 n+1 + z p n+2 + z q n+3 .
According to Theorem 2.2 and h-cobordism theorem [START_REF] Smale | On the structure of 5-manifolds[END_REF], the Brieskorn knot K Pp,q is spherical. Note that K P and K Pp,q are respectively of dimension 2n + 1 and 2n + 3.

Levine's complete invariant set of cobordant isometric structures.

We refer to [START_REF] Levine | Invariants of knot cobordism[END_REF] for all results of this subsection.

In order to study C n the cobordism group ok knotted n-dimensional spheres in codimension two, J. Levine introduced isometric structures <, >, T where <, > is a non degenerate symmetric bilinear form on a finite dimensional vector space V over Q, and T is an isometry of V . Recall that the group law of C n is the connected sum ; two such knots K 0 and K 1 are cobordant if the oriented connected sum K 0 # -K 1 is null-cobordant, i.e. cobordant to the trivial embedding, and this only happens when the associated Seifert forms are Witt-equivalent.

Following [START_REF] Levine | Invariants of knot cobordism[END_REF], an isometric structure <, >, T is null-cobordant when V contains a totally isotropic subspace, invariant under T , of half dimension of V . Moreover, two isometric structures <, >, T and <, > ′ , T ′ are cobordant if the orthogonal sum <, >, T ⊥ -<, > ′ , T ′ is null-cobordant. As in the case of spherical knots, this equivalence relation gives an abelian group of cobordism classes of isometric structures.

Let ∆ T (t) be the characteristic polynomial of T . Levine proved that the group of cobordism classes of isometric structures <, >, T satisfying ∆ T (1)∆ T (-1) = 0 is isomorphic to the Witt-equivalence group of matrices A satisfying (A -t A)(A + t A) is non-singular. With this isomorphism we associate the isometric structure A + t A, -A -1 t A to a square matrix A over Q such that (A -t A)(A + t A) is non-singular. This isomorphism allows to study Witt-equivalence of matrices in terms of cobordism classes of isometric structures.

Let <, >, T an isometric over Q. Let Λ = Q[t, t -1 ] be the ring of Laurent polynomials over Q. We consider V the vector space on which <, > and T are defined as a Λ-module, defining the action of t by T . Let ∆ T (t) = r i=1 λ i (t) ei be the factorisation of ∆ T (t) with irreducible factors over Q. To each irreducible factor λ i we define V λi = Ker λ i (t) N for N a large integer, such a V λi is called a primary component of V . Moreover, V is the direct

sum V = r i=1 V λi .
Let λ(t) be a symmetric 1 irreducible factor of ∆ T , then Levine defined

(1) ε λ <, >, T equals to the exponent of λ(t) in ∆ T (t) mod 2.

(2) σ λ <, >, T equals to the signature of the restriction of <, > to V λ over R.

(

) µ λ <, >, T = -1, -1 r(r+3 2 det(<, >), -1 r S <, > 3
where <, > is of rank 2r, the Hilbert symbol for <, > over R is denoted by , and S <, > is the Hasse symbol over Q We have the following theorem.

Theorem 2.3 ([11]

). Two isometric sturctures α and β are in the same cobordism class if and only if ε λ (α) = ε λ (β), σ λ (α) = σ λ (β) and µ λ (α) = µ λ (β) for all λ(t) for which these invariants are defined.

Results

3.1.

Cobordism classes of Brieskorn knots. First, we prove the following proposition.

Proposition 3.1. Let K P and K Q be two Brieskorn knots associated with two polynomials P and Q with n+1 variables. If there exists two distinct prime numbers p and q such that the spherical Brieskorn knots K Pp,q and K Qp,q are not cobordant, then K P and K Q are not cobordant.

Proof. Since the polynomial P p,q is obtained from P by adding a two variables polynomial of the form z p i + z q i+1 , then according to K. Sakamoto [START_REF] Sakamoto | The Seifert matrices of Milnor fiberings defined by holomorphic functions[END_REF] we know that the Seifert form of the Brieskorn knot K Pp,q is obtained by the tensor product of the Seifert form of K P with a square matrix A p,q . By Durfee [START_REF] Durfee | Fibered knots and algebraic singularities[END_REF] we know that A p,q is a matrix with only 0 and ±1 coefficents which are only determined by p and q.

If two Brieskorn knots K P and K Q are cobordant, then they have witt-equivalent Seifert forms A KP and A KQ defined on Z-modules G P and G Q . Let M , a submodule of G P ⊕ G Q , be a metabolizer for A KP ⊕ -A KQ and let d be the dimension of the square matrix A p,q . Then the submodule

d i=1 M i of d i=1 G P,i ⊕ G Q,i , where M i is a copy of M and G P,i ⊕ G Q,i is a copy of G P ⊕ G Q , is a metabolizer for A KP ⊗ A p,q ⊕ -A KQ ⊗ A p,q .
Hence the two Seifert forms A KP ⊗ A p,q and A KQ ⊗ A p,q are Witt-equivalent. Since the Brieskorn knots K Pp,q and K Qp,q are spherical, we know that they are cobordant. We have proved the proposition by contraposition. are cobordant if and only if the Brieskorn knots K P+ and K Q+ associated to the polynomials

P + (z) = z 2 n+2 + n+1 j=1 z aj j and Q + (z) = z 2 n+2 + n+1 j=1 z bj j are cobordant.
Proof. According to [START_REF] Sakamoto | The Seifert matrices of Milnor fiberings defined by holomorphic functions[END_REF] the knots K P and K P+ have same Seifert forms and the same is true for K Q and K Q+ . Since the cobordism class of fibered knots is completely determined by the algebraic cobordism class of its Seifert form (see [START_REF] Blanloeil | A theory of cobordism for non-spherical links[END_REF]) we have that K P and K Q have algebraically cobordant Seifert forms if and only if the same holds for K P+ and K Q+ . Then, the Brieskorn knots K P and K Q are cobordant if and only if the Brieskorn knots K P+ and K Q+ are cobordant.

Recall that if A is a Seifert matrix of a Brieskorn knots K P associated with a Brieskorn polynomial

P (z) = z a1 1 +z a2 2 +• • •+z an+1 n+1
, then its Alexander polynomials is defined by ∆ K (t) = det t A + (-1) n t A . Since Brieskorn knots are fibered knots, we know that the Alexander polynomial is the characteristic polynomial of the monodromy. Proof. By the monodromy theorem proved by E. Brieskorn [START_REF] Brieskorn | Die Monodromie der isolierten Singularitäten von Hyperflächen[END_REF] the Alexander polynomial of Brieskron knots are some products of cyclotomic polynomials. Moreover, Alexander polynomials of Brieskorn knots has been computed by F. Pham [START_REF] Pham | Formules de Picard-Lefschetz généralisées et ramification des intégrales[END_REF] and E. Brieskorn [START_REF] Brieskorn | Beispiele zue Diffenrentialtopologie von Singularitaten[END_REF]. More precisely, let P be the polynomial P (z) =

∆ KP (t) = 0<i k <a k t -ζ i1 a1 . • • • .ζ in an ( * )
where ζ a k = e 2πi a k . Let K P and K Q be two Brieskorn knots with distinct irreducible factors in their Alexander polynomial, then they have roots aver C which are distinct. The Brieskorn knots K Pp,q and K Qp,q are defined with the same integers p and q which are distinct prime numbers and both are coprime with all exponents of P and Q. Then, according to ( * ) the Alexander polynomials of K Pp,q and K Qp,q have distinct irreducible factors since the do not have the same complex roots. Moreover, proposition 2.2 implies that K Pp,q and K Qp,q are spherical Brieskorn knots.

According to [START_REF] Durfee | Fibered knots and algebraic singularities[END_REF], if A is the Seifert form associated to the Milnor fibration, then its intersection form is defined by the relation S = A + (-1) n t A and its monodromy is defined by the relation h = (-1) n-1 A -1 t A. Moreover by proposition 3.2 one can suppose that (-1) n = 1. Hence we have that S, h is an isometric structure on H n (F, Q) where F is the Milnor fiber of the isolated singularity at 0 of a Brieskorn polynomial, S is the intersection form and h is the monodromy. On top of that, up to invertible element, the characteristic polynomial of h is the Alexander polynomial defined using the Seifert form which is a product of symmetric cyclotomic polynomials.

Finally, according to theorem 2.3, the two spherical Brieskorn knots K Pp,q and K Qp,q cannot be cobordant since they do not have the same list of invariants.

In [START_REF] Blanloeil | Cobordism of algebraic knots defined by Brieskorn polynomials[END_REF] we proved that cobordant Brieskorn knots must have same exponents, up to order, provided no exponent is a multiple of another one. Now, we have the following result which is an immediate corollary of proposition 3.4.

Corollary 3.5. Let K P and K Q be the Brieskorn knots associated to the polynomials

P (z) = n+1 j=1 z aj j and Q(z) = n+1 j=1 z bj j .
Let P P and P Q be the sets of distinct irreducible factors of each polynomial P and Q. If P P =P Q , then the knots K P and K Q are not cobordant.

Proof. Set ζ a k = e 2πi a k , as before we have the following factorizations

∆ KP (t) = 0<i k <a k t -ζ i1 a1 . • • • .ζ in an and ∆ KQ (t) = 0<i k <b k t -ζ i1 b1 . • • • .ζ in bn .
When P P =P Q the polynomials ∆ KP and ∆ KQ admit distinct complex roots.

Hence they do not have the same irreducible factors over Q. By proposition 3.4 the knots K P and K Q are not cobordant.

In the following, for each Brieskorn knot, we will define a list of integers which only depends on its exponents. Since these integers are related to the factorization of the Alexander polynomial of as a product of cyclotomic polynomials, then we will get an invariant of cobordism classes of Brieskorn knots.

In the following, when α and β are two integers, we denote by α ∧ β the greatest common divisor of α and β and we denote by [α, β] the lowest common multiple of α and β.

Let k and d be two integers. Set µ k (d) be the greatest divisor of k ∧ d which is coprime with k k∧d . Then, we associate to the couple of integer (k, d) a set of integers defined as follows , for all distinct prime numbers p and q which are both coprime to each a 1 , . . . , a n+1 we define the set Ξ KP ,p,q as follows.

Ψ k (d) =        [k,d] l , if l is a divisor of k µ k (d) k∧d \ {k} if µ k (d) ≤ 2 [k,d] l , if l is a divisor of k µ k (d)
(1) Ξ 0 KP ,p,q = {pq} (2) for i = 1 to n + 1 we set Ξ i KP ,p,q = Ψ αj (a i ) | α j ∈ Ξ i-1 KP ,p,q

(3) Ξ KP ,p,q = Ξ n+1 KP ,p,q

We will prove that Ξ KP ,p,q is invariant in a cobordism class of Brieskorn knots.

Proposition 3.7. Let K P be a Brieskorn knot associated with the polynomial

P (z) = n+1 j=1 z aj j .
Let p and q be two distinct prime numbers which are both coprime to a 1 , . . . , a n+1 , then the set of integers Ξ KP ,p,q is an invariant of its cobordism class Proof. When P (z) is a Brieskorn polynomial with z = (z 1 , z 2 , . . . , z n+1 ) and n ≥ 1 ; I. Savel'ev [START_REF] Savel'ev | Cyclotomic polynomials and singularities of complex hypersurfaces (Russian) Uspekhi Mat. Nauk[END_REF] compute the Alexander polynomial of Q(z, z n+2 ) = P (z) + z d n+2 where d is an integer. Let Φ n (t) be the n-th cyclotomic polynomial. If the Alexander polynomial of P (z) is

∆ P (t) = N l=1 Φ k l (t) τ l , with τ l > 0, then ∆ Q (t) = N l=1 ν|λ k l ,d Φ [k l ,d] ν (t) ϕ(µ k l (d)) Φ k l (t) τ l (⋆)
where λ k l ,d = dµ k l (d) k l ∧d . We see that {Φ k1 , . . . , Φ kN } is the set of irreducible factors of ∆ P (t) and Φ Ψα j (d) | α j ∈ {k 1 , . . . , k N } is the set of irreducible factors of ∆ Q (t).

According to [START_REF] Brieskorn | Beispiele zue Diffenrentialtopologie von Singularitaten[END_REF], the Alexander polynomial of the Brieskorn knot K x p +y q is ∆ p,q (t) = (t pq r -1) r (t -1) (t p -1)(t q -1) where r = p ∧ q. If p and q are distinct prime numbers, then we have

∆ p,q (t) = Φ pq (t)Φ p (t)Φ q (t)Φ 1 (t) 2 Φ p (t)Φ q (t)Φ 1 (t) 2 = Φ pq (t).
When µ k l (d) ≤ 2, then ϕ(µ k l (d)) equals 1 and Φ k l is no longer a factor of ∆ Q in (⋆). Hence Ξ KP ,p,q is the set of irreducible factors of ∆ Pp,q . According to proposition 3.4, when p and q are two distinct prime numbers which are both coprime to each a 1 , . . . , a n+1 , the set Ξ KP ,p,q is an invariant of the cobordism class of K P .

Remark 3.8. The last proposition is just a reformulation of proposition 3.4. But it gives a computable list of integers which is an invariant of the cobordism class of a Brieskorn knot.

3.2.

Examples. Recall that Alexander polynomials of cobordant knots must satisfy a Fox-Milnor relation. In [START_REF] Blanloeil | Cobordism of fibered knots and related topics[END_REF] we only use this property to determine if knots cannot be cobordant. Moreover, in the same paper, example 3. Hence, the polynomials ∆ K f and ∆ Kg have not the same set of complex roots. By proposition 3.4 we see that in this case the Brieskorn knots K f and K g are not cobordant.

3.3. Conjecture. We found more examples of Brieskorn knots for which distinct exponents imply distinct cobordism classes. Moreover, according to proposition 3.7 it seems difficult to find examples of Brieskorn knots which are cobordant and have distinct exponents up to order. Hence, we formulate the following conjecture.

Conjecture 3.9. The Brieskorn knots associated to the polynomials Remark that if this conjecture is true, then the multiplicity of a Brieskorn knot will be an invariant of its cobordism class.

P (z) =

Proposition 3 . 2 .

 32 The Brieskorn knots K P and K Q associated to the polynomials e., such that λ(t) = ±t deg(λ) λ(t -1 ).

Remark 3 . 3 .Proposition 3 . 4 .

 3334 The knots K P and K P+ have same Seifert forms, hence they have same Alexander polynomials. The set of irreducible factors of the Alexander polynomial over Q of a Brieskorn knot is an invariant of its cobordism class.

  Brieskorn gave the following factorization over C for the alexander polynomial of the Brieskorn knots K P

k∧d else Definition 3 . 6 .

 36 When K P is a Brieskorn knot associated with the polynomialP (z) =n+1 j=1 z aj j

3 n- 3 + z 6 n- 2 + z 6 n- 1 + z 6 n + z 6 n+1

 33626166 [START_REF] Kato | A classification of simple spinnable structures on a 1-connected Alexander manifold[END_REF] gave the examples of Brieskorn knots K f and K g where n > 3, p 1 , p 2 , . . . , p n-3 ≥ 2 andf (z) = z p1 1 + z p2 2 + • • • + z pn-3 n-3 + z 8 n-2 + z 8 n-1 + z 4 n + z 4 n+1 and g(z) = z p1 1 + z p2 2 + • • • + z pn-for wich it was unknown if they are in the same cobordism class since the product of the Alexander polynomials fulfills a Fox-Milnor relation. With proposition 3.4 we have a new method to determine if such Brieskorn knots lies in the same cobordism class or not. More precisely, if 3 is coprime with all p 1 , . . . , p n-3 , then the set of prime factors of the exponents of the polynomials f and g are distinct.Recall that when P (z) = n+1 j=1 z aj j we have ∆ KP (t) = 0<i k <a k t -ζ i1 a1 . • • • .ζ in an .

  and only if a j = b j , j = 1, 2, . . . , n + 1, up to order.


Maurice Bernadet

teaching and research institutions in France or abroad, or from public or private research centers.

Les transports terrestres de marchandises aux Pays-Bas. Actualisation