AA STUDY ON PREFIXES OF c INVARIANTS
KAREN YEATS
Abstract.
This paper begins by reviewing recent progress that has been made by taking acombinatorial perspective on the c invariant, an arithmetic graph invariant with connectionsto Feynman integrals. Then it proceeds to report on some recent calculations of c invariantsfor two families of circulant graphs at small primes. These calculations support the ideathat all possible finite sequences appear as initial segments of c invariants, in contrast totheir apparent sparsity on small graphs. Introduction
The c invariant is an arithmetic graph invariant introduced by Schnetz in [11] in orderto better understand certain Feynman integrals. The c invariant sees aspects of the sameunderlying geometry that the Feynman period sees [3, 4] and consequently the c invariantcan predict things about what classes of numbers can show up in a given Feynman period.See Subsection 3.3 for some further comments in this direction. In the following graphs willbe assumed to be connected unless otherwise mentioned. For a graph G , the c invariant of G , c ( G ) is a sequence of numbers indexed by primes c ( G ) = ( c (2)2 ( G ) , c (3)2 ( G ) , c (5)2 ( G ) , c (7)2 ( G ) , c (11)2 ( G ) , . . . )with each c ( p )2 ( G ) ∈ Z /p Z . The definition of c ( p )2 ( G ) is given in the next section. As we willsee, c ( q )2 can be defined for prime powers, not just primes, but we will stick to primes herein.Previous work of Brown and Schnetz [4] calculated c invariants for graphs up to 10 loops atsmall primes (up to the first 100 primes in some cases). These calculations uncovered manyinteresting patterns, most notably coefficient sequences of q -expansions of modular forms.In [14], the author described a new, more graphical technique for calculating c invariants.With Wesley Chorney, this technique was expanded in [8]. For this technique the prime p is fixed but there is a finite algorithm to calculate the c invariant at p for all members ofa recursively constructed class of graphs. The loop orders of the graphs in these classes areunbounded, so these techniques let us calculate c invariants at all loop orders albeit withfixed p and only for certain families.The c invariant is also believed to have certain symmetries corresponding to symmetriesof the Feynman period. One of these is called completion symmetry and was conjecturedby Brown and Schnetz in 2010 in [3]. This conjecture has turned out to be quite difficult.This combinatorial perspective on the c invariant is used in [15] to prove one special caseof the conjecture. An overview of the results of [14], [8], and [15] is given below in Section 3 Thanks to Oliver Schnetz for goading me into doing the prefix calculation, and for his continued excitementabout the c invariant. Thanks to Iain Crump and Freddy Cachazo for discussions. Thanks to NSERC andthe Humboldt foundation for support. Thanks to the organizers of the Algebraic Combinatorics, Resurgence,Moulds and Applications (CARMA) conference for their excellent conference. a r X i v : . [ m a t h . C O ] S e p long with an outlook for this approach and connections to topics of particular interest tothe CARMA conference.The explicit calculations in [14] and [8] were done by hand and so involve only p = 2and relatively simple classes of graphs. The complexity of the calculation as a function of p is quite bad, and it also grows depending on the graph class. Nonetheless, by computersome new progress is possible, though only a little, the results of which are reported on inSection 4. These new computations are particularly interesting because they let us probethe behaviour of the c invariant at all loop orders, giving a rather different impression thanprevious exhaustive computations at fixed loop orders.2. Set up
Let G be a 4-regular graph. The graph resulting from removing any one vertex of G (andits adjacent edges) is called a decompletion of G , and G is called the completion of any ofits decompletions. In general there may be many non-isomorphic decompletions of a graph.For any graph H (but of primary interest is the case when H is a decompletion of a 4-regular graph) associate an indeterminate a e for each edge e and define the (dual) Kirchhoffpolynomial or first Symanzik polynomial of G to beΨ H = (cid:88) T (cid:89) e (cid:54)∈ T a e where the sum runs over all spanning trees T of H . For example the Kirchhoff polynomialof a 3-cycle with edge variables a, b, c is a + b + c .Now we can define the c invariant. Given a polynomial f with integer coefficients, write[ f ] q for the number of F q -rational points on the affine variety defined by f = 0 (with f firstreduced to F q ). Our polynomials f will always come from a graph in one way or another,and so the affine space in which they are to be taken will always be of dimension the numberof edges of the graph. Definition 2.1.
Suppose H has at least 3 vertices, then c ( q )2 ( H ) = [Ψ H ] q q mod q. See [3] for a proof that this is well-defined. Note that in [3] they have the condition thatthe dimension of the cycle space of H is at most two less than the number of edges of H ,however using Euler’s formula this condition is equivalent to H having at least 3 vertices. Inwhat follows we will restrict to c invariants at primes p , not more general prime powers, sincethis is computationally accessible and corresponds to what has been calculated elsewhere [4].We can also view Ψ H as a determinant in the following way. Choose an arbitrary orderfor the edges and the vertices of H and choose an arbitrary orientation for the edges of H .Let E be the signed incidence matrix of H with one row removed and let Λ be the diagonalmatrix of the edge variables of H . Let M = (cid:20) Λ E t − E (cid:21) . Then by the matrix-tree theorem det( M ) = Ψ H . See [2] Proposition 21 or [13] for details.) The matrix M behaves much like the Laplacianmatrix of a graph with variables included and with one matching row and column removed,but the pieces which make it up are expanded out by blocks, so call M the expanded Laplacian of H .As well as det( M ), minors of M are useful. For I and J sets of edge indices (or sets ofedges; with the edge order fixed, we need not distinguish between an edge and its index)let M ( I, J ) be the expanded Laplacian with rows indexed by elements of I removed andcolumns indexed by elements of J removed. In [2] Brown gave the following definition. Definition 2.2.
Let I , J , and K be sets of edge indices with | I | = | J | . Define Ψ I,JG,K = det( M ( I, J )) | a e =0 for e ∈ K When K = ∅ we will simply leave it out. Brown called these polynomials
Dodgson polynomials . They satisfy many relations, see[2]. Different choices in the construction of M may change the overall sign of a Dodgsonpolynomial, but since we will be concerned with counting zeros of these polynomials theoverall sign is of no interest.The combinatorial perspective on the c invariant comes from now taking a different viewpoint. Instead of thinking about the c invariant in terms of polynomials and point counts,we want to think about it in terms of set partitions of subsets of vertices and spanning forestpolynomials. We need a few lemmas to get to this reinterpretation. Lemma 2.3 (Lemma 24 of [3] along with inclusion-exclusion) . Suppose | E ( H ) | ≤ | V ( H ) | . Let i, j, k be distinct edge indices of H and let p be a prime. Then c ( p )2 ( H ) = − [Ψ ik,jkH Ψ i,jH,k ] p mod p. Note that if H is a decompletion of 4-regular graph with at least 2 vertices then it satisfiesthe hypotheses of the previous lemma. This lemma is useful because we no longer have todivide by p but rather directly count points modulo p . Combined with the next lemma, weno longer need to count points at all. Lemma 2.4.
Let F be a polynomial of degree N in N variables, x , . . . , x N , with integercoefficients. The coefficient of x p − · · · x p − N in F p − is [ F ] p modulo p . This lemma is a corollary of one of the standard proofs of the Chevalley-Warning theorem,see section 2 of [1].Together these two lemmas tell us that to calculate the c invariant we only need tounderstand the coefficient of (cid:89) ≤ (cid:96) ≤| E ( H ) | (cid:96) (cid:54) = i,j,k a p − (cid:96) in (cid:16) Ψ ik,jkH Ψ i,jH,k (cid:17) p − . We can make this yet more combinatorial by reinterpreting these Dodgson polynomials interms of sums over spanning forests.
Definition 2.5.
Let P be a set partition of a subset of the vertices of H . Define Φ PH = (cid:88) F (cid:89) e (cid:54)∈ F a e here the sum runs over spanning forests F of H with a bijection between the trees of F andthe parts of P where each vertex in a part lies in its corresponding tree. Trees consisting ofisolated vertices are allowed. Call these polynomials spanning forest polynomials . Dodgson polynomials can always berewritten in terms of spanning forest polynomials. This is a manifestation of the all-minorsmatrix tree theorem [7]; the following form is convenient for the present purposes.
Proposition 2.6 (Proposition 12 from [6]) . Let I , J , and K be sets of edge indices with | I | = | J | . Then Ψ I,JH,K = (cid:88) ± Φ PH \ ( I ∪ J ∪ K ) where the sum runs over all set partitions P of the end points of edges of ( I ∪ J ∪ K ) \ ( I ∩ J ) such that all the forests corresponding to P become spanning trees in both G \ I/ ( J ∪ K ) and G \ J/ ( I ∪ K ) . The signs in the sum can be determined, see Proposition 16 of [6]. All that we will needis that if the set partitions are of the form { a, b } , { c, d } and { a, c } , { b, d } then they appearwith opposite sign, see Corollary 17 of [6].The two lemmas told us to calculate the coefficient of (cid:81) (cid:96) (cid:54) = i,j,k a p − (cid:96) in (cid:16) Ψ ik,jkH Ψ i,jH,k (cid:17) p − modulo p . Now, we can interpret the two Dodgson polynomials as signed sums of spanningforest polynomials, and so we are interested in the coefficient of (cid:81) (cid:96) (cid:54) = i,j,k a p − (cid:96) in each ofcertain products of 2 p − p then calculates the c invariant. Notice that each spanningforest polynomial is, by construction, linear in each edge variable. So taking this coefficientamounts to determining which of the variables to assign to each polynomial in the productand taking the resulting monomial from each polynomial. If a particular variable is assignedto a particular spanning forest polynomial then we are restricting ourselves to the spanningforests in that polynomial which do not use that edge. If a particular variable is not assignedto a particular spanning forest polynomial then we are restricting ourselves to the spanningforests in that polynomial which do use that edge, or equivalently, to spanning forests in thegraph with that edge removed with one more tree than before made from breaking up thetree which originally used that edge.Given a subset S of the edges of a graph H and a spanning forest polynomial Φ PH , an assignment of the edges of S to H is the polynomial resulting from a choice for each edge of S to either assign it or not assign it to Φ PH . That is, given a choice of a subset S (cid:48) ⊆ S , theresulting polynomial is the coefficient of (cid:81) e ∈ S (cid:48) a e in Φ PH | a e =0 ,e ∈ S − S (cid:48) . This new polynomial isitself a sum of spanning forest polynomials, as the following lemma describes. Lemma 2.7 (Lemma 5.2 from [8]) . Given a spanning forest polynomial on a graph H and aset S ⊆ E ( H ) , any assignment of the edges of S yields a sum of spanning forest polynomialson the graph H − S . Furthermore, the vertices involved in the set partitions defining thenew spanning forest polynomials involve only vertices already in partition for the originalpolynomial along with vertices incident to S . Note that some set partitions may give impossibilities, in this case the spanning forestpolynomial is an empty sum, and so is 0. Also, we can discard isolated vertices from H − S as by connectivity they must each be in their own part of every set partition and so theycontribute no information. . Past applications of this method
Completion.
One of the important reasons to study the c invariant is to better un-derstand the Feynman period. For a graph H the Feynman period, in affine form, is definedto be P H = (cid:90) a i ≥ (cid:81) da e Ψ H (cid:12)(cid:12)(cid:12)(cid:12) a =1 . This is a residue of the Feynman integral in parametric form which is independent of kine-matical parameters. If H comes from a four regular graph K with one vertex removed and K is internally 6-edge-connected, that is any way of removing fewer than six edges of K either leaves the graph connected or disconnects only an isolated vertex, then the integralconverges. The Feynman period is known to have four important symmetries, most of whichwere long known in physics, and which can be found in a form as we will use them in [10].The symmetry we will focus on is the completion symmetry, namely if H and H are bothdecompletions of the same 4-regular graph G then P H = P H . This is proved for the periodby moving to momentum space and inverting the variables, see [10].The Feynman period is controlled by the geometry of the denominator of the integral, thatis by the geometry of the variety Ψ H = 0. The c invariant is accessing this geometry from adifferent direction, by counting rational points in that same variety over various finite fields.Thus we should expect that they are saying something about each other. However, the c invariant is only seeing part of the geometric structure, not the whole thing. As it turns outthe c invariant has been very useful in predicting properties of the period, most notably if c ( p )2 = 0 for all p then we expect the period to have less than maximal transcendental weightfor the size of the graph. We also expect that if two graphs have the same period then theyshould have the same c invariant. The converse is certainly not true, with c ( p )2 = 0 for all p being a good example. One consequence of this is that the c invariant should have all thesymmetries that the period does.In particular Brown and Schnetz, [3], conjectured that if G is a connected 4-regular graphand v and w are vertices of G then c ( p )2 ( G − v ) = c ( p )2 ( G − w ) for all primes p .The main result of [15] is a very special case of this conjecture and the first major progresstowards the conjecture. Theorem 3.1 (Theorem 1.2 of [15]) . Let G be a connected 4-regular graph with an oddnumber of vertices. Let v and w be vertices of G . Then c (2)2 ( G − v ) = c (2)2 ( G − w ) . The approach of [15] is combinatorial following the set up described in Section 2. For p = 2this is particularly simple since we are looking for the coefficient of (cid:81) (cid:96) (cid:54) = i,j,k a (cid:96) in Ψ ik,jkH Ψ i,jH,k modulo 2; that is, we need to determine the parity of the number of edge assignmentscompatible with Ψ ik,jkH Ψ i,jH,k , and each edge assignment assigns exactly one copy of each edge,dividing them between the two polynomials, so we are counting certain edge bipartitions.Further, it suffices to prove the result for v and w adjacent, leaving the remaining vertexbetween v and w i, j, k to be the threeincident edges to this vertex. This means that for both G − v and G − w we are down toconsidering spanning forest polynomials on the graph G − { v, w } . Further, then, in eitherdecompletion Ψ ik,jk = Ψ G −{ v,w } , and so we only need to count the parity of the number ofedge bipartitions such that one part is a spanning tree of G − { v, w } and the other part is a panning forest compatible with Ψ i,jk ; which spanning forests these are is the only thing thatchanges between G − v and G − w .What we need is for the number of these edge bipartitions to have the same parity between G − v and G − w . This takes some work; the bipartitions fall into two classes dependingon how the vertex partitions giving the spanning forest polynomials divide the vertices. Forsome of the cases, we can define a fixed point free involution showing that the set of theseedge bipartitions is even. For other cases, no such construction was evident, and so instead amore complicated construction was used involving an auxiliary graph related to the spanningtree graph. What was needed to finish the proof was that this auxiliary graph has an evennumber of vertices, as the vertices correspond to the edge bipartitions in the remainingcases. When G has an odd number of vertices, all the vertices of the auxiliary graph haveodd degree and so the auxiliary graph as a whole has an even number of vertices. When G has an even number of vertices then we do not have this parity restriction on the vertexdegrees in the auxiliary graph, and so the proof does not extend directly.3.2. Circulants and toroidal grids.
Another use of the approach described in Section 2is to calculate c invariants for fixed p but for whole families of graphs. This contrasts with[4] where Brown and Schnetz fix the graph and calculate the c invariant for many primes.They do this systematically for small graphs, collecting many interesting results.Taking the fixed p and graph family approach, in [14] and [8] we have both specific andgeneral results. For the specific results we need to define some classes of graphs. Definition 3.2.
The circulant graph C n ( i , i , . . . , i k ) is the graph on n vertices with an edgebetween vertices i and j if and only if i − j = i (cid:96) mod n or j − i = i (cid:96) mod n for some ≤ (cid:96) ≤ k . We will be interested in certain 4-regular circulant graphs. Every 4-regular circulant graphcan be written as C n ( i, j ) with i (cid:54) = n − j and i, j (cid:54) = n/
2. Note that each decompletion of agiven circulant graph is isomorphic, so we will use the notation (cid:101) G for the decompletion of G , which is well-defined in the case that G is a circulant or other vertex-transitive graph.Circulant graphs are an interesting class of graphs for questions related to c or to Feynmanperiods because they include both the simplest non-trivial family, namely the zigzags [5],which are the C n (1 , Definition 3.3.
The nonskew toroidal grid indexed by ( k, and (0 , m ) is the Cartesianproduct of a cycle on k vertices and a cycle on m vertices. These are nonskew toroidal grids as we can also define more general toroidal grids wherewe begin with the Cartesian product of a cycle on k vertices and a path with m edges,making a finite cylindrical grid, and then identify the top and bottom k -cycles, potentiallywith an offset. In the case where there is a nonzero offset (cid:96) then the result is a skew toroidalgrid and if gcd( m, (cid:96) ) = 1 then it is isomorphic to the circulant graph C km ( (cid:96), m )The author ([14]) and the author with Wesley Chorney ([8]) proved the following explicitresults heorem 3.4. • c (2)2 ( (cid:102) C n (1 , n mod 2 for n ≥ ( [14] Proposition 4.1). • c (2)2 ( (cid:94) C k +2 (1 , k )) = 0 mod 2 for k ≥ ( [14] Proposition 7.1). • Let G be a nonskew toroidal grid indexed by ( N, , (0 , m ) with at least N vertices.Then c (2)2 ( (cid:101) G ) = 0 ( [8] Proposition 3.2)
In [8] we also prove that two other families have c (2)2 = 0.The technique behind these results is an algorithm which applies to any recursively con-structible family of graphs with the property that Lemma 2.3 eventually applies to membersof the family. The notion of recursively constructible family is due to Noy and Rib´o, see[9] section 2. For its use in this c context see [8] section 5. We do not need the precisedefinition here, but the idea is as follows.Roughly, recursively constructible families consist of graphs made with some initial pieceand then a chain of repeated structures and then a cap which may link the last piece of thechain back to the initial piece. Let { H n } be the graphs of this family with n the length ofthe chain. Let H (cid:48) n be H n with the edges of the cap deleted. Fix a prime p . Using Lemma 2.3with three edges in the cap and using Lemma 2.7 to assign all other edges in the cap, we cancalculate c ( p )2 ( H n ) by taking the coefficient of (cid:81) e ∈ H (cid:48) n a p − e in some sum of products of 2 p − H (cid:48) n where the partitions only use vertices in the final piece ofthe chain and in the initial piece. Note that there are only a finite number of spanning forestpolynomials of this form and so only a finite number of products of 2 p − H (cid:48) n − so what we obtain from Lemma 2.7 is a sum of products of 2 p − H (cid:48) n − . What this tells us is that we can calculate the coefficient of (cid:81) e ∈ H (cid:48) n a p − e in any of these products as a sum of coefficient of (cid:81) e ∈ H (cid:48) n − a p − e in products for H (cid:48) n − . So what we have is a system of first order linear recurrences. The system itself can becomputed in a finite amount of time (at worst all possible products of 2 p − H (cid:48) . Then it remains to solve thesystem modulo p . This algorithm is due to the author with Wesley Chorney and gives thefollowing theorem Theorem 3.5 ([8], Theorem 5.3) . Let G n be a recursively constructible family of graphs with | V ( G n ) | = | E ( G n ) | + 2 for n sufficiently large. The c invariant for any fixed prime p canbe calculated using these methods in a finite amount of time for all graphs of the family. The special case of this algorithm for the circulant graphs used in Section 4 was alreadyin [14].As a piece of mathematics the existence of this algorithm is very nice as it says that it istheoretically possible to rigorously compute c invariants for entire families of graphs, albeitonly for fixed p . In practice it is not so nice. The specific results listed above were obtainedby applying this method on the specific examples using hand computations. Note that westuck to the particularly nice case of p = 2 for the explicit results. The complexity of themethod as a function of p is very poor. The complexity of the method also grows quickly inthe number of vertices which H (cid:48) n − shares with the next piece of chain. · ·· · · ab c d ef · · ·· · · a b c d e f Figure 1. (cid:102) C n (1 ,
3) (left) and (cid:102) C n (2 ,
3) (right).3.3.
Comments and outlook from past results.
The c invariant began as a tool linkingthe Feynman period, which is in some sense a physical object, to the arithmetic of theKirchhoff variety. These come from different aspects of the geometry of the Kirchhoff variety.This is why some arithmetic questions on the kinds of numbers appearing in the Feynmanintegrals and their transcendental properties have a fairly tight link to the c invariant andits properties. With the methods surveyed above another major perspective and toolset,that of combinatorics, is brought to the study of the c invariant and progress can be made.The completion result, partial though it is, is a testament to the power of algebraic and enu-merative combinatorics. By doing some counting using classic enumerative tools like fixed-point-free involutions progress was made where more algebro-geometric tools were stuck. Inthe end both areas are important as the reduction to the counting problem is fundamentallyarithmetic in nature. This interplay between areas is part of what makes these problemsenjoyable and is central to the aims of the CARMA conference and the broader CARMAproject.We can also ask: what does it mean that all nonskew toroidal grids have c (2)2 = 0? Whenthe c -invariant is 0 for all p then we expect a drop in transcendental weight; it is too muchto hope that this is what is happening here, though to clarify this one of the most interestingexplicit calculations to do now would be p = 3 for some nonskew toroidal grids. More likelythe symmetries of the toroidal grids only force that c (2)2 = 0. It is not clear what this meansgeometrically, nor, in the other direction, what other graphs behave in this way, though someother examples are known.Towards the future, the obstacles to extending the partial completion result do not appearinsurmountable and are the subject of ongoing research. Getting beyond p = 2 will involveseeing how the larger number of possibilities collect into sets of size p . We can also try tocollect more data from the family approach. At this point in the study of the c invariantmore data nearly invariably shows new patterns and raises new questions. The remainder ofthis document is a report on a computerization of the circulant c calculation in the (cid:102) C n (1 , (cid:102) C n (2 ,
3) cases and a discussion of some of the perhaps surprising things which can beseen in this data. 4.
Computerized circulant c computations The goal now is to implement the algorithm described in Subsection 3.2 in as practical amanner as possible on certain families of circulant graphs. rom now on we will only consider the families (cid:102) C n (1 ,
3) and (cid:102) C n (2 , (cid:102) C n (1 ,
3) we process the edges labelled 1,2,3, and 4 in Figure 1 and for (cid:102) C n (2 ,
3) weprocess the edges labelled 1,2, and 3. This can be done by hand with the lemmas of theprevious section, and in fact these calculations were done in [14]. As it turns out, for (cid:102) C n (1 , c and d comes for free, giving c ( p )2 ( (cid:102) C n (1 , (cid:89) e ∈ H n a p − e ] (cid:16) Φ { a,f } , { b } , { e } H n Ψ H n (cid:17) p − mod p,c ( p )2 ( (cid:102) C n (2 , (cid:89) e ∈ K n a p − e ] (cid:16) Φ { b,e } , { c } , { d } K n (cid:16) Φ { c,d } , { b,e } H n − Φ { c,b } , { d,e } H n (cid:17)(cid:17) p − mod p where here square brackets are the combinatorialist’s notation for coefficient of , where H n is (cid:102) C n (1 ,
3) with vertices c and d and their incident edges deleted, and where K n is (cid:102) C n (2 , C n (1 ,
3) and Section 5 of [14] for C n (2 , e for H n and d for K n . Note that for the products of spanning forest polynomialscalculated in the previous paragraph, at most the first three and last three vertices of thegraph are used in the set partitions. In view of Lemma 2.7, this remains true (now in H n − or K n − ) after assigning the edges of one piece of the chain. Thus the products of spanningforest polynomials which could appear are any products of 2 p − c itself and addingto the list as needed. The number of products, call this N , which were necessary for theprogram in each case computed is shown in Table 1. For the actual edge assignments, we areonly reducing two edges at a time, so for each polynomial in the product there are only fourpossibilities, both edges are in, both edges are out, the one edge alone is in, or the other edgealone is in. How each of these affects the vertex partition is coded for each case and then theedge assignment calculation simply comes down to looping over the ways of assigning edgesbetween the polynomials. The outcome of this step of the algorithm is the N × N coefficientmatrix of the system of recurrences.The next step of the algorithm is to calculate the initial conditions. We need N initialconditions. Each of these is calculated on the smallest graph of the family. The smallest H n has 5 vertices and 6 edges; the smallest K n has 5 vertices and 5 edges. This is the casewhere the last of the first three vertices is the same as the first of the last three vertices. For Specifically, if F is a polynomial and m is a monomial then [ m ] F is the coefficient of m in F . (1 , p N C (2 , p N Table 1.
Number of products of spanning forest polynomials ( N ) necessary.both the minimal H n and minimal K n cases, how each possible spanning forest partitionsthe vertices is precomputed, then for a given product of spanning forest polynomials eachpartition is compared to the precomputed list to get the count. The outcome of this step isa vector of length N .The final step is then to solve this system of linear recurrences with these initial conditions.This is not done algebraically because the matrices get very large. Rather, the system issimply iterated. The sequence of c invariants at p is then the sequence of first entries ofthese iterated vectors in the H n case and a weighted sum of the first p + 1 entries in the K n case. Eventually these sequences seem to begin to repeat. It turns out that the vector atthe point where the c sequence first repeats is not yet equal to the initial condition vector.Rather, it takes multiple iterations of the repeating block of the c sequence before the vectormatches the initial condition vector. Call the period of repetition of the c sequence the c period and call the period of repetition of the vector the vector period . Verifying that thevector agrees with the initial condition vector at the vector period proves that the systemrepeats with this period and then checking the c sequence breaks into blocks according tothe c period within one vector period proves that the c sequence repeats with the observed c period as well.Unfortunately, the vector periods are quite large making them computationally problem-atic. The easiest and most naive way to compute the vector period is simply to iterate thesystem and compare the resulting vector with the initial condition vector. This process is notguaranteed to terminate as there could be transient behaviour in the early iterations. Thesimplest example of such transients would be if there was a row of all 0s but a nonzero initialcondition in that location, but longer transients are also possible. All we are guaranteedtheoretically is that, by finiteness of the field, at some point the result of an iteration agreeswith some past iteration. In principle this is also true of the c itself, but in practice the c sequence displays periodic behaviour beginning at the very first value. A less naive wayto compute the vector period would compare it with past values after each c period. Thedownside of this approach is that it is slower and uses more memory.Attempts were made to compute the vector periods for p = 5 for C (1 ,
3) and for p = 3 for C (2 , p = 5 and C (1 ,
3) the less naive computation method was used but ran out of Note the different use of the term period from earlier sections: for the remainder of the paper, periodwill mean period of repetition of a sequence (1 , p c period vector period2 2 43 36 590405 37207 134064 C (2 , p c period vector period2 7 563 4356 Table 2.
Periods of repetition for the system of recurrences. The c periodsare proven when the vector period is listed and are otherwise empirical.memory after three weeks; the vector period in this case exceeds 153844320. For p = 3 and C (2 ,
3) the more naive computation run for a month suggests that the vector period exceeds4614354360, though it remains possible that initial transient behaviour simply means thatthe vector period is not obtainable by the naive method. An attempt to compute the vectorperiod for p = 7 for C (1 ,
3) was not made because iterating the system until the pointwhere the c sequence appeared to repeat took over a month and the vector period wouldbe expected to be many times this.The periods are given for each computed case in Table 2. Note that the c periods arenot proved, only empirically observed, for p = 5 and p = 7 in the C (1 ,
3) case, nor for p = 3for the C (2 ,
3) case. However, the empirical evidence is quite strong. For p = 5 and C (1 , c sequence repeated exactly 41356 times before the computation was killed.For p = 3 and C (2 , c sequence repeated exactly 1059310 times beforethis document was submitted.Observe that the c periods are all much smaller than one would naively expect given thesizes of the matrices and also considerably smaller than the vector periods. This means thatthere is a substantial amount of structure which this method does not capture. The ratiobetween the c periods and the vector periods gives the first hint of where this additionalstructure may reside. Looking at the vectors after each c period, the first many entriesagree while some later entries do not. The system is built so that for each entry of the vectorthe corresponding product of spanning forest polynomials does appear in the construction,but the behaviour of the vector and the c sequence indicates that various values of the laterentries are equivalent for the c calculation. Playing around with the coefficient matrix inthe C (1 , p = 2 case, suggests that some block decomposition might be possible in orderto explain at least some of the discrepancy between the c period and the vector period.Unfortunately, the structure is not clear for the p = 3 coefficient matrix. Understandingthis redundancy should be the next step for both the theoretical and practical take on thisalgorithm.Without such additional reductions, the computations presented here exhaust what wecan do for (cid:102) C n (1 ,
3) and (cid:102) C n (2 , p = 7 computation for (cid:102) C n (1 ,
3) took 100GB of RAMand took several months to run on a University of Waterloo server. Even then the systemwas only iterated until the c sequence appeared to repeat. Specifically, the system was refix C (1 , C (2 , Table 3.
Frequencies for prefixes ( c (2)2 , c (3)2 ).iterated until the first 1351 entries repeated and prior to this there was no reoccurrence ofan initial segment of length greater than 6. Consequently, p = 11 will be outside the rangeof practical computation. The p = 5 case for (cid:102) C n (2 ,
3) was attempted but was killed as itexceeded 400GB of RAM; N had already surpassed 10 million and rough heuristics based onhow N grew during the other computations suggests that the final N for p = 5 for (cid:102) C n (2 , c invariants themselves are presented. All the sequences have beenverified for small values of n by Oliver Schnetz using different techniques. c (2)2 ( (cid:102) C n (1 , , ∗ c (3)2 ( (cid:102) C n (1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ∗ c (2)2 ( (cid:102) C n (2 , , , , , , , ∗ The remaining computed c sequences, c (5)2 ( (cid:102) C n (1 , c (7)2 ( (cid:102) C n (1 , c (3)2 ( (cid:102) C n (2 , (cid:102) C n (1 ,
3) begin at n = 9 which corresponds to 7 loopdecompleted graphs. The sequences for (cid:102) C n (2 ,
3) begin at n = 7 which corresponds to 5loop decompleted graphs. The ∗ indicates to repeat the sequence indefinitely. For example c (2)2 ( (cid:102) C n (1 , , ∗ means that c (2)2 ( (cid:102) C (1 , c (2)2 ( (cid:103) C (1 , c (2)2 ( (cid:103) C (1 , c (2)2 ( (cid:102) C n (1 ,
3) was computed by hand in [14] and that the three sequencesdisplayed above are proved while the remaining three are only empirically observed.5.
Discussion
This data is interesting and important because it lets us probe c invariants at all looporders, albeit only on these two families of graphs and only for a very few initial primes.One particularly interesting question is which finite sequences occur as initial sequencesof c invariants ( c (2)2 ( H ) , c (3)2 ( H ) , c (5)2 ( H ) , c (7)2 ( H )) in this data. This is easy to tally, we justtake the least common multiple of the periods to get the period for the initial segments so farand then count how many of each occur. To begin with consider the prefix ( c (2)2 , c (3)2 ). Thereare 6 possible prefixes and the distributions for each family are shown in Table 3. Note thatfor (cid:102) C n (1 ,
3) it is uniform, while for (cid:102) C n (2 ,
3) the difference between the largest and smallestcounts is less than 6% of the total number. c o un t Figure 2.
Number of occurrences of length 4 prefixesFor (cid:102) C n (1 ,
3) we can also consider the prefixes of length 3 and 4. For the prefixes of length3 the period for the prefix is 11160. The numbers of occurrences of the prefixes are between350 and 393; the difference between these is less than 0 .
4% of the total. The mean is 372.Performing the same calculations on the prefixes of length 4. The period for the prefix is20779920. The counts all lie between 87514 and 110213; the difference between these isslightly over 0 .
1% of the total. The mean of the counts is 98952. The number of occurrencesfor the prefixes of length 4 are plotted in Figure 2.From this we see that the distribution of the frequencies of the different prefixes is quite flatand does not seem to be getting any less flat as we take larger prefixes. In particular everyprefix occurs and there is no indication that this will change as we move to longer prefixes.This is an interesting and perhaps unexpected observation because looking at small graphsleaves the impression that only rather few finite sequences occur as prefixes of c invariants.This data, which can probe all loop orders, suggests quite the opposite: perhaps all finitesequences can occur as prefixes of c invariants.Geometrically, this says that the possible geometries for Kirchhoff varieties should not beexpected to be sparse among all possible geometries, rather, at least as far as finite prefixes an see, it looks like everything can happen, though it may take very large loop order to getthere. References [1] James Ax. Zeroes of polynomials over finite fields.
Amer. J. Math. , 86(2):255–261, 1964.[2] Francis Brown. On the periods of some Feynman integrals. arXiv:0910.0114.[3] Francis Brown and Oliver Schnetz. A K3 in φ . Duke Math J. , 161(10):1817–1862, 2012. arXiv:1006.4064.[4] Francis Brown and Oliver Schnetz. Modular forms in quantum field theory.
Communications in NumberTheory and Physics , 7(2):293 – 325, 2013. arXiv:1304.5342.[5] Francis Brown and Oliver Schnetz. Single-valued multiple polylogarithms and a proof of the zigzagconjecture.
Journal of Number Theory , 148:478–506, 2015. arXiv:1208.1890.[6] Francis Brown and Karen Yeats. Spanning forest polynomials and the transcendental weight of Feynmangraphs.
Commun. Math. Phys. , 301(2):357–382, 2011. arXiv:0910.5429.[7] Seth Chaiken. A combinatorial proof of the all minors matrix tree theorem.
SIAM J. Alg. Disc. Meth. ,3(3):319–329, 1982.[8] Wesley Chorney and Karen Yeats. c invariants of recursive families of graphs. Ann. Inst. Henri Poincar´eComb. Phys. Interact. , (to appear). arXiv:1701.01208.[9] Marc Noy and Ares Rib´o. Recursively constructible families of graphs.
Adv. Appl. Math. , 32(1):350–363,2004.[10] Oliver Schnetz. Quantum periods: A census of φ -transcendentals. Communications in Number Theoryand Physics , 4(1):1–48, 2010. arXiv:0801.2856.[11] Oliver Schnetz. Quantum field theory over F q . Elec. J. Combin. , 18, 2011. arXiv:0909.0905.[12] N. J. A. Sloane. The on-line encyclopedia of integer sequences. , 2008.[13] Aleks Vlasev and Karen Yeats. A four-vertex, quadratic, spanning forest polynomial identity.
Electron.J. Linear Alg. , 23:923–941, 2012. arXiv:1106.2869.[14] Karen Yeats. A few c invariants of circulant graphs. Commun. Number Theory Phys. , 10(1):63–86,2016. arXiv:1507.06974.[15] Karen Yeats. A special case of completion invariance for the c invariant of a graph. Canad. Math. J. ,70(6):1416–1435, 2018. http://dx.doi.org/10.4153/CJM-2018-006-5. Also arXiv:1706.08857.,70(6):1416–1435, 2018. http://dx.doi.org/10.4153/CJM-2018-006-5. Also arXiv:1706.08857.