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On the Counting of Colored Tangles

The connection between matrix integrals and links is used to define matrix models which count alternating tangles in which each closed loop is weighted with a factor n, i.e. may be regarded as decorated with n possible colors. For n=2, the corresponding matrix integral is that recently solved in the study of the random lattice six-vertex model. The generating function of alternating 2-color tangles is provided in terms of elliptic functions, expanded to 16-th order (16 crossings) and its asymptotic behavior is given.


Introduction
The problem of counting topologically distinct knots remains a challenging one: see [1] for a review and for a report on recent advances. It was recently noticed [2] that combinatorial methods developed in quantum field theory, namely Feynman diagrams applied to matrix integrals, may provide a new way to count knots. In particular, the counting of alternating tangles, which had been achieved in [3], was reproduced. This counting, however, is not capable of discriminating between objects with different numbers of connected components: in the usual terminology, it gives the number of links rather than that of knots. In the present article, we reexamine this question and show how the introduction of a number n of possible "colors" for the knotted loops would solve this question, and how this may be formulated in terms of a matrix integral. For n = 2, this integral is equivalent to one recently studied in detail and computed in the framework of the random lattice 6-vertex model [4][5]. We thus carry out the explicit counting of alternating 2-color tangles: their generating function is the solution of coupled equations involving elliptic functions. We are able to give the 13 first terms of its expansion and its asymptotic behavior. We conclude with more conjectural considerations on the number of (2-color alternating) links.

A matrix model for colored links
We want to consider a model which describes alternating links in which each of the intertwined loops can have n different colors. This can be achieved via a large N × N matrix integral [2]. As a first attempt, let us consider the following integral Such an integral has a g-series expansion ("perturbative expansion") which admits a graphical representation in terms of Feynman diagrams. In the large N limit, only planar diagrams survive: see for example [6] for a review. The integral (2.1) has an O(n)-invariance where the M a form the vector representation of O(n). The Feynman diagram expansion of F (n, g) generates planar diagrams with four-legged vertices and colored edges such that colors cross each other at each vertex ( Fig. 1).
To each planar diagram we can associate an alternating link diagram (see e.g. [7], page 21) by following colored loops as they cross other loops and choosing alternatingly under-and over-crossings. This can be carried out in a consistent way throughout the whole diagram. We have thus generated alternating link diagrams with n colors. Diagrams with different numbers of connected components can now be distinguished by the n dependence of their weight: indeed, to a diagram with k connected components is associated a factor n k . We can therefore write where F k (g) is the sum over alternating link diagrams with exactly k intertwined loops.
Note that if one can define F (n, g) for non-integer n and in particular in a neighborhood of 0, one has access to the individual contributions F k (g) since they form the small n expansion of F (n, g). As usual in O(n) vector models, one can perform (at least formally) an analytic continuation in n by a Hubbard-Stratonovitch transformation. Unfortunately such a transformation breaks planarity of the diagrams and is not suitable for our purposes.
However, another Hubbard-Stratonovitch transformation exists for the particular values n = ±2 (besides n = 1, of course) which preserves planarity, as will be explained later.
We can also define correlation functions in the model. There is only one 2-point function, where a is fixed, 1 ≤ a ≤ n. Here the brackets · refer to the normalized average with the exponential weight of (2.1). Due to O(n)-invariance, there are two independent 4-point functions. We consider connected 4-point functions only; we choose where a and b are fixed and distinct. They have the following interpretation:  We must now proceed as in [2]. Our goal is 1) to count only diagrams which are prime and reduced, and 2) to count as a single contribution diagrams which correspond to the same link. First it is necessary to take care of non-prime and non-reduced diagrams. This is achieved by introducing an extra parameter t in the action: and choosing t as a function of g in such a way that We have the obvious scaling property G(t, g) = 1 t G(1, g/t 2 ), which means that given the two-point function of the original model G(1, g) ≡ G(g), t(g) is the solution of the equation: t(g) = G(g/t 2 (g)) . (2.8) We have similar scaling properties for the higher correlation functions; in particular, The next step is to remove the over-counting of links due to the fact that several diagrams can correspond to a single link. According to the Tait flyping conjecture (proven in [8]), two reduced alternating diagrams are equivalent if and only if they are related by a sequence of flypes (Fig. 3). In order to take into account the flyping equivalence, we must now consider, as advocated in [9], the most general O(n)-invariant model with quartic

T T
There is a new type of vertex which allows loops to "avoid" each other. The appearance of two types of vertices can be understood as follows: the unwanted (i.e. overcounted) tangle diagrams due to the flyping equivalence can be of either type 1 or type 2, and we must introduce "counterterms" of both types to cancel them. To pursue the analogy with renormalization, we can rephrase this by saying that it is only the "renormalized" coupling constants which must be of the form (g, 0), but in order to reach this point we must consider more general "bare" coupling constants (g 1 , g 2 ).
We impose again the condition From now on t will be assumed to be fixed by (2.10), and we shall be concerned with the correlation functions Γ i ≡ Γ i (t(g 1 , g 2 ), g 1 , g 2 ), which are the generating functions of diagrams of type i with four external legs and two types of vertices weighted by g 1 and g 2 respectively.
As in [2], we make use of the concepts of two-particle irreducibility. Following the language of field theory, we recall that a four-legged diagram is two-particle-irreducible (2PI) if cutting any two distinct propagators leaves it connected; otherwise it is two-particlereducible (2PR). Also, we shall consider skeleton diagrams, which must be "dressed" to recover ordinary diagrams. This will be implicit in the following and we refer the reader to [2] for details.
Let us define D i (resp. H i , V i ) to be the generating functions of 2PI (resp. 2PI in the horizontal, vertical channel) four-legged diagrams of type i. Note that H 1 = V 1 , but since the defining property of diagrams of type 2 is not invariant by rotation of By decomposing a general diagram Γ 1 according to the number of times it is reducible in the horizontal channel (Fig. 4), we find the following formulae: .  This can be simplified by introducing the combinations Γ ± = Γ 2 ± Γ 1 and similarly for H ± . We find: Note that these equations are independent of n.
It is not as obvious how to decompose diagrams of type 2 using the vertical channel.
It is simpler to proceed as follows: define Γ 0 to be generating function of diagrams with four external legs such that the color of the two left outgoing strings is free, whereas the color of the two right outgoing strings is fixed and equal; and similarly H 0 , V 0 , D 0 . We have the formulae We can now proceed to decompose Γ 0 in the horizontal channel. It is easy to convince oneself that the simple formula holds.
The three equations (2.11) and (2.13) determine H 1 = V 1 , H 2 and V 2 as functions of Γ 1 and Γ 2 . 1 More explicitly, one should invert them to and take appropriate linear combinations. Furthermore, noting that any diagram with four external legs and four-legged vertices must be irreducible in one of the two channels, we have which means that we have managed to express the D i in terms of the Γ i .
The flype equivalence does not affect 2PI skeleton diagrams; this means in practive that the expression of D ′ i = D i − g i as a function of the Γ i is left unchanged by the removal of the overcounting of flype equivalent diagrams. Note that the D i [Γ 1 , Γ 2 ] have a simple expression given by Eqs. (2.14)-(2.15), but the g i [Γ 1 , Γ 2 ] are non-trivial functions 1 If one sets n = 0, two equations become identical and one has to include the derivative of (2.13) with respect to n: which can only be obtained by actually solving the matrix model (2.9) (or some equivalent procedure). For example, perturbatively we find: (Fig. 5) where the order of truncation is dictated by the rule: 2 Γ 1 ∼ g, Γ 2 ∼ g 2 . The data D ′ i [Γ 1 , Γ 2 ] is all we need from the generalized matrix model (2.9), and we can now come back to the problem of tangle diagrams, with a single coupling constant g weighting a simple crossing.
We want to redo the decompositions of a general diagram in the horizontal channel, but this time taking into account the flyping equivalence. This forces us to distinguish between simple crossings and other diagrams, that is to use the primed objects defined bỹ Here, as in [2], the tilde on one of the symbols Γ, H or V denotes generating functions of flype equivalence classes of 2PI skeleton diagrams.
We invert these relations toH and express the 2PI functions Finally, the generating functionsΓ i (g) are given by the implicit equations comes from the solution of the matrix model, andD ′ i (g) is given by Eqs. (2.20)-(2.21) withΓ i =Γ i (g).

The n = 2 case: a model of oriented links
Let us now carry out explicitly the procedure outlined in the previous section, for a value of n corresponding to a solved matrix model. The case n = 1 has already been considered in [2]; let us now set n = 2. The partition function is conveniently rewritten in terms of a complex matrix X = t 2 (M 1 + iM 2 ): where we have absorbed t in the coupling constants We recognize in (3.2) the partition function of the six-vertex model on random dynamical lattices, which has recently been solved [4,5]. In the 6 vertex formulation it is particularly clear that there is a Hubbard-Stratonovitch transformation which preserves planarity; 3 introducing the notation c = 2b cos θ, with θ ∈ [0, π[ in the regime of interest to us, we have where A is a hermitean matrix. (Here and in the following integral (3.4), we omit an overall constant factor, irrelevant in the computation of correlation functions.) We have rewritten our model as a model of oriented loops which are not the intersecting loops we started from, since they can only avoid each other, see Fig. 6 b). This may seem unnatural, but is what allows to solve exactly the model, since we can now integrate over the original matrix X, shift the matrix A: and do a saddle point analysis of the eigenvalues of A. This is the basis of the analytic solutions [4,5] of the model.
It is natural to redefine the two independent 4-point functions to be i.e. they are characterized by the position of the ingoing/outgoing arrows on the external legs. They are related to the 4-point functions defined earlier by: as is clear diagrammatically.
Let us summarize the combinatorial relations of the previous section in the case n = 2.
First, the relations which determine the corresponding 2PI correlation functions D b and D c can be obtained either by direct diagrammatic arguments or by setting n = 2 in the general formulae (2.11)-(2.13). We find: Similarly, the modified relations which take into account the flyping equivalence read: (we have reintroduced the renormalized coupling constant g corresponding to a simple crossing, We shall now use the solution of the matrix model (3.2) [4,5] to show how to extract the functionsΓ perturbatively at an arbitrary order, and to find their large order behavior.

Perturbative expansion of the off-critical solution.
In [5], the large N saddle point density of eigenvalues of A is explicitly constructed in terms of elliptic functions. More precisely if we define the resolvent where the average is with respect to the measure in (3.4), and then we have the following expression for J using an elliptic parametrization u: where H is the Jacobi theta function and A, B, u ∞ a 0 are constants which depend on b 0 and θ.
If we consider the small b, c perturbative expansion of the correlation functions of the model, we are in the region where the elliptic nome q is close to zero, and the elliptic functions can be expressed to a given order in q in terms of trigonometric functions. We can therefore write every quantity as a power series in q with coefficients dependent on θ.
From the 1/a term in the a → ∞ expansion of J(a) we can extract W 1 = lim N→∞ 1 N trA , and from there the two-point function using the formula We can then go back to the free energy by integrating once (at fixed θ) and differentiate again to get the 4-point functions: The rescaling which allows to set G = 1 simply amounts to writing that the coupling (3.14) Then we have We must apply formulae (3.6) to evaluate D ′ b and D ′ c . Finally, we must consider q and θ as power series in the renormalized coupling constant g and solve perturbatively in g the equations (3.7).
Also, these data are compatible with those of Tab. 1.  In [4], the model (3.2) is analyzed in the critical regime, that is when one encounters the first singularity of the free energy (or of the correlation functions) by analytic continuation from the gaussian model. This singularity -or more precisely, the line of singularities in the (b 0 , c 0 ) plane -is what determines the large order behavior of the correlation functions, and ultimately of the functionsΓ .
We shall now summarize some of the relevant results from [4]. If the parameter θ is fixed and the coupling constant b 0 is increased, for any value of θ ∈ [0, π[ there will be a critical value b ⋆ 0 (θ) for which the free energy becomes singular: .
This gives us access to H = ∂ ∂θ F |b 0 fixed at criticality. One finds: One must then use again the formulae (3.14)-(3.15) and (3.6) to find the values of Γ b , Γ c , c on the critical line as a function of θ, and finally solve (3.7) for g and θ. This is a set of 2 complicated coupled equations, 4 and we can only solve them numerically. We find the following numerical values: The value (3.20b) is non-universal, and only this explicit calculation could give us access to it. It is related to the leading behavior of the coefficients of the power seriesΓ b (g) andΓ c (g), i.e. of the numbers of oriented prime alternating tangles. On the other hand, the subleading behavior is universal; it is characteristic of c = 1 conformal field theories coupled to gravity, which exhibit zero "string susceptibility" with logarithmic corrections [4,5]. Therefore we can state that ifΓ b (g) = γ p g p , then γ p p→∞ ∼ const g −p c p −2 (log p) −1 (3.21) and similarly forΓ c . The constant 1/g c = 6.28329764 . . . is slightly less than the corresponding constant 16/(π(π − 4) 2 ) = 6.91167 . . . [9] for oriented alternating link or tangle diagrams (without taking into account the flype equivalence); and should also be compared to the numbers obtained for n = 1: (101 + √ 21001)/40 = 6.147930 . . . and 27/4 = 6.75 with and without the flype equivalence, respectively. The fact that the n = 1 and n = 2 results are fairly close shows in particular that the entropy generated by tangles with large numbers of connected components is small.
As a final note, we can make some slightly conjectural statements on the number of oriented prime alternating links. It is clear that if we close a tangle by pasting a simple crossing to its four external legs, we obtain a link; if the tangle had k closed loops, then the number of connected components of the resulting link is simply k + 2 for a tangle of type 1 and k + 1 for a tangle of type 2. Inversely, from a link with p crossings one can produce 4p tangles by removing one of its vertices and fixing the circular permutation of the four legs. Even though we cannot use this fact to correctly count links (one cannot make it a one-to-one correspondence), by assuming that most links have low symmetry, we have the approximate relation f p ≈ γ p−1 /p satisfied by the number f p of links with p crossings. Therefore we conjecture that f p p→∞ ∼ const g −p c p −3 (log p) −1 .