New Invariant Quantity to Measure the Entanglement in the Braids

In this work, we demonstrate that the integral formula for a generalised Sato-Levine invariant is consistent in certain situations with Evans and Berger’s formula for the fourth-order winding number. Also, we found that, in principle, one can derive analogous high-order winding numbers from which one can calculate the entanglement of braids. The winding number for the Brunnian 4-braid is calculated algebraically using the cup product on the cohomology of a ﬁnite regular CW-space, which is the complement R 3 \B 4 .


Introduction
A braid (or n-braid) is a collection of n strands stretching between 2 parallel planes that twist and cross. One example is the pure braid in which the strands start and end with the same permutation, see for instance, a 3-braid in Fig. 1 (left). Identifying the two parallel planes of this 3-braid yield a link, Fig. 1 (right).
Braids and links are used to model many physical systems that include entanglement in their force field lines (integral curves), see for example the coronal loops above the Sun's photosphere Fig. 1 (left). In this situation, the magnetic reconnection occurs due to the entanglement of the magnetic field lines and then the magnetic topology changes. This process leads to a conversion of the magnetic energy into kinetic energy and a heating of the plasma.
Understanding of magnetic reconnection is based on the Sweet-Parker model [1]. In a magnetic field, the connectivity of the field lines can occur only at null points, where the magnetic field vanishes. This model gives a rough scaling law predicting the rate of reconnection at the null point. On the other hand, there is no requirement that reconnection be limited to such locations; indeed, numerical experiments show the development of turbulent reconnection in magnetic fields with no null points.
In astrophysical environments, the magnetic fields are usually inherently disordered, being characterised by field lines that are tangled with one another 1 (right), [2,3], typical picture for braided magnetic field is shown in Fig.   Figure 2. Coronal loops above the Sun's photosphere (left), some typical field lines are braided and contained within a tubular domain (right).
Winding numbers are used easily to do many topological calculations for example braid invariants which have applications in fluid mechanics, molecular biology and astrophysics [4].
The idea of introducing integrals of magnetic fields in the form of the Massey products first appeared in [5], and has been developed in [6,7,8]. Different types of higher-order invariants have been introduced, for instance, a fourth-order linking was introduced by [9] and another integral formula for the generalised Sato-Levine invariant was introduced by [10]. The permutation problem is a serious obstacle for calculating thirdorder linking numbers. A requirement for higher-order winding numbers is to avoid such problems [11]. In this paper, we are showing that the Evans-Berger and Akhmetev's formulas coincide in certain cases. Also, we will introduce another formula for the fourth-order linking number in analogy to the formula of lower-order. This formula is introduced in section (7), it allows to measure the linking of the field lines of a magnetic field (e.g. magnetic braid) when the lower order linking numbers are trivial. This formula is tested on a pure braid and also the result consist with that obtained algebraically in section 8.

Linking of two tubes
Suppose we have two closed curves α 1 and α 2 . These curves are parametrised by arclengths t and s, respectively. The positions along α 1 and α 2 are given by x 1 (t) and x 2 (s). Let r = x 1 − x 2 be the relative position vector. The linking number between α 1 and α 2 , introduced by Gauss (1867) has the following representation, Gauss emphasised that L 12 has a value equal to the signed number of crossing in a projection of the link. This Gauss integral is invariant under deformations of the two curves in which intersecting not allowed. Now, in order to define the linking of two flux tubes we take two thin flux tubes U 1 and U 2 with unit fluxes. The curves α 1 and α 2 are enclosed in U 1 and U 2 , respectively. The vector potential A 2 due to U 2 is defined according to the Ampér's theorem in the following form where the vector functions x 1 (t) and x 2 (s) define points lying on the curve α 1 and α 2 , which are the axes of the thin flux tubes U 1 and U 2 , respectively. Now assume we have unit fluxes, Gauss linking integral can be written in the following way (3) Let B 1 and B 2 be two divergence-free magnetic field and V be a simply connected volume such that the boundary ∂V is a socalled flux surface, i.e. B 1 · n| ∂V = B 2 · n| ∂V = 0. The crosshelicity of these magnetic fields is defined in the following form The cross-helicity measures the cross-linkage of flux between B 1 and B 2 . It can be written in a form to include the linking number. If we apply it to a system of two isolated thin flux tubes U 1 and U 2 which correspond to the fields B 1 and B 2 , we obtain For this reason it is known as a second-order topological invariant (i.e. it is quadratic in magnetic flux φ i ). Eq. (3) can be reformulated as a volume integral over the cross-helicity as follows Now, define a new field M 12 = A 1 × A 2 , which is divergencefree everywhere in the spaceŨ = R 3 \ ∪ 2 i=1 U i , and apply the divergence theorem with the help of the vector identity [28], Then Eq. (3) becomes 3. Third-Order Linking of Three Tubes The Gauss integral cannot distinguish between the Whitehead link or the Borromean rings on one hand and the trivial link on the other hand. So, this integral is the lowest order used to measure the entanglement of curves and interlocking of flux tubes. Based on the Massey triple Evans and Berger (1992) constructed a third-order link integral which can be used to show that the Borromean rings are not unlinked. This integral is one of a whole hierarchy of linking integrals. Now, suppose that the closed curves α i , i = 1..3 of the Borromean rings are enclosed in tori U i with vector potentials A i of magnetic fields B i such that these fields vanish outside the toroidal volume U i , and let U be the complement of the subspace ∪ 3 i=1 U i ⊂ R 3 . From algebraic topology, use the fact 'the homology H 1 (U i ; R) and the cohomology H 1 (U i ; R) are isomorphic', by using the Alexander Duality Theorem 1 , these are isomorphic to the homology and cohomology of the complement of U i , and consequently, any basis of H 1 (U i ; R) and any basis of H 1 (R 3 \U i ; R) could be isomorphic. For this reason each tube U i corresponds to a unique cohomology class in H 1 (R 3 \U i ; R). Also, we can identify the vector potential A i with a differential one-form A i such that dA i = B i , where B i is a differential two-form corresponding to the magnetic field B i in the tube U i , and d is the exterior derivative. Define the two-form M i j which are the dual of the divergence-free vector fields M i j = A i × A j (defined recently). In this sense, the Gauss linking integral will take the following form Using stokes' Theorem the last integral becomes The linking numbers L i j (i j) of the curves of the Borromean rings vanish. The one-forms A q represent a cohomology classes in H 1 (U ; R) and the two-form M pq represent a cohomology classes in H 2 (U , R). We can add the two-form φ i | U j B j to the form M i j inside the tube U j and take away the form φ j | U i B i from M i j inside the tube U i , where φ i | U j is a scalar potential satisfying A j = dφ i | U j . This is an approach by Berger to make the M i j closed everywhere, and this allows us to find a potential N i j such that dN i j = M i j . Now the modified form is 1 Alexander Duality Theorem applies to the homology theory properties of the complement of a subspace X p in e.g. Euclidean space R n , then H q (R n \X) and H n−q−1 (X) are isomorphic.
The Poincaré lemma is helpful to show that closed forms represent cohomology classes, i.e. the closed forms are exact at least locally. Here we remind the reader of the lemma : Theorem 1 (Poincaré lemma). Suppose X is a smooth manifold, Ω k (X) is the set of all smooth differential k-form on X, and suppose ω is a closed form in Ω k (X) for some k > 0. Then for every x ∈ X, there is a neighbourhood U ⊂ X and a (k-1)-form η ∈ Ω k−1 (X), such that dη = ι * ω, where ι is the inclusion ι : U → X, (this is a proper Poincaré lemma) [12]. Moreover, if X is contractible, this η exists globally; there exists a (k-1)-form η ∈ Ω k−1 (X) such that dη = ω.
With the help of the Poincaré lemma, there exists a oneform N i j such that dN i j = M i j . In the space ∪ 3 i=1 U i , the Massey product associated with the cohomology classes is defined as follows [13]. Let and define a two-form M i jk by This form represents an element in the cohomology class in H 2 (U , R) [13,14]. The Massey product is introduced firstly by the following volume integral, which is known as the thirdorder linking integral expanding the integrand term in the last equation yields By considering that the flux tube is thin with unit flux, the last integral can be reformulated as a line integral (see e.g. [15], p. 14-15) to take the following form We will show how the line integral of the Massey product Eq. (14) and the third-order winding number are related to each other in the particular case in which the braid is closed (periodic) and then can be considered as a link.
The third-order winding number is an invariant quantity up to smooth deformation and since every pure braid can be combed, 3 it is enough to consider a combed braid only. Here, we take a pure 3-braid as an example, comb it such that two strands, α j and α k say, are fixed and the other, α i , winds around them. For this braid the third-order integral, Eq. (15), becomes This is achieved by considering that α i and α j are complex functions. The terms r i j and θ i j are the modulus and argument of α i j = α i − α j . Now, we want to show how Eq. (16) can match (14). For this purpose, the integral of the one-form A k along the path α i equal the flux induced by B k weighted by the winding angle θ ik Now differentiate with respect to τ to obtain Since A j on U i = dφ j | U i , then φ j | U i can be written as follows Also, the two-form M jk on U i is just Since, z i j = r i j exp(−i θ ij ) and z ki = r ki exp(−i θ ki ) and since, d ln z i j ∧ d ln z ki = 0, then we can get the following relation Now, by using the above identity in the Eq. (20), the following form can be used as a potential for the two-form M jk Assuming that the tubes U i are filled with unit longitudinal fluxes, Eq. (14) takes a new form equivalent to Eq. (16) after defining its terms as in Eqs. (17), (19) and (22).

Evans-Berger's Formula for the Fourth-Order Linking Number
Evans and Berger (1992) [9] have used the same idea of construction of the third-order winding to build a formula for the fourth-order linking number. They formulated the Massey quadruple in vector notation so as to construct the fourth-order link invariant for magnetic flux tubes in the form of a volume integral. They used the example of a link including four curves, say α i , i = 1..4, which are enclosed in thin toroidal volumes U i and each tube filled with divergence-free magnetic fields B i . These fields point in the same direction as the axial curves α i and do not cross the boundary (i.e. B i ·n = 0). For each field a flux in the directions of the central line Φ i . The magnetic fields have vector potentials A i (i.e. ∇ × A i = B i ). For this link the second and third order linkings vanish (i.e. L i j = L i jk = 0), and since we are working in a simply connected space, there exists a one-form N i jk everywhere such that where M i jk defined as follows One can check that the two-form M i jk is closed, (i.e. dM i jk = 0). The scalar potentials ω jk | U i and ω i j | U k are defined in the way that the following condition hold The integrand of the fourth-order linking integral, known as "Massey quadruple product", is defined as then the fourth-order linking can be written as a surface integral in the following way Now, by applying stokes' theorem, Eq. (27) can be reformulated as a volume integral as follows

Integral formula for a generalised Sato-Levine invariant
A fourth-order integral W has been proposed by [10], for a pair of divergence-free magnetic fields B i and B j respectively localised in two oriented tubes U i and U j in R 3 . The integral W is invariant up to deformations of the configuration space, its value is preserved in the motion of tubes in an ideal medium. This integral is a generalisation of the Sato-Levine invariant which is defined for two tubes with zero linking number [16]. The Sato-Levine invariant β has been discovered independently by Jerome Levine, also, [17] and [18] have also studied this invariant. 4 The invariant is defined for a semiboundary link. Such a link has the property that every a pair of its components has a zero linking number, and the invariant is an integer which can be calculated by studying the intersections of Seifert surfaces 1 of the components of the link. It's proper definition is just for a pair of two linked curves [16]. Later, [19] generalised it for a higher order. Akhmetiev start with a particular case of two tubes with zero linking number to construct W [20], and later they improved the formula to be applicable in more general cases, and they described the dependence between the integral W and the generalised Sato-Levine invariant.
In the next section, we will review the construction of the integral formula (W) of Akhmetiev and Ruzmaikin. We plan to show how this formula is related to the fourth-order invariant of Evans and Berger, so we will restrict ourselves to the case in which the linking numbers is zero.

Formula of the invariant W
The following construction is taken from [10] to derive the fourth-order integral W, but here it is restricted to case for which the linking numbers vanish. We denote by U i and U j the magnetic flux tubes with fixed orientation of their axial lines α i and α j respectively, equipped by magnetic fields B i and B j respectively. The vector potentials of these fields are A i and A j respectively. Now, consider the restriction A j | U i and define a multivalued function τ i : U i → R as follows where x 0 is a fixed point in U i , and s is a path connecting the point x 0 and x in the torus U i . The function τ i can be defined by A j | U i say λ j . In this case, the following relations hold for the tubes U i and U j respectively, Now consider the vector field associated with a certain closed 2-differential form if the axial lines of the tubes U i and U j have zero linking number The choosing for the potential ϕ suitably to makes F divergencefree accordingly, since There exists a vector potential G for the field F, (i.e. ∇×G = F).
If we suppose the requirement ∇ × G = 0 with a suitable boundary condition, then the vector field G is uniquely determined by F. Akhmetiev and Kunakovskaya have introduced the integral W by the formula

The Consistency of Evans-Berger and Akhmetev's Formulas
In this part, we will show how the formula of the integral formula of Akhmetev is consistent with the fourth order-linking of Evans and Berger after grouping the strands in pairs. Now we will start with expanding the Formula (28) by calculating the term dM i jkl Now we will identify strand i and k and also strand j and l. We can do this by renaming k = i and l = j and then do the calculation We can ignore the sixth term in Eq. (32) because we will integrate over the volume U i where B j vanishes, and by summing the third and fourth terms, we will get Let us denote by K the last two terms in Eq. (33) and using definition of M i jk , as well as Eqs. (23) and (14), K becomes (since, the terms ω ji | U i and ω i j | U i represent the third-order integrals of only two tubes, they are equal to zero). Now, by using Eqs. (10) and (23), one can verify that the second term in Eq. (33) equal to −φ 2 j | U i B i ∧ A i , then we will get Evans and Berger's formula for the fourth-order linking number becomes Now, by restricting the integration in Akhmetev's formula, Eq. (31), to one flux tube, e.g. U i , and translate Eq. (36) into vectorial language, the resultant forms are consistent.

A Higher Analog of the Winding Number
Accounting of the higher-order winding integral is eased by using differential forms. In the Brunnian 4-braid, all the second and third-order winding numbers vanish, see where C 0 = u + (− cos(s), sin(s), s) , sin(s − π), s) . . .
This also can be obtained from κ where a(s) = κ(a(0)). The other strings can be written in a similar way. Observe that all the lower winding numbers (λ i j and Ψ i jk ) vanish. The following one-form is the analogy of ψ abc (third-order winding number), and it is a closed form as shown below Since the integrand ψ i jk is a closed form, then all the terms which include dψ i jk vanish. Expanding the last equation and using ω i j = ω ji and Arnold's identity, then we will get The real part of the integral provides a fourth-order winding number, which is an invariant quantity up to deformation measuring the linkage of the four strands of the braid κ.
In the following part, we will test this formula for a braid corresponding to a well known link "Brunnian ring". Now, we will apply the above formula for the braid B 4 in Figure 3. First, we label the strings of the colours (red, blue, green and black) by a, b, c and d, respectively.
(42) Figure 4 shows the fourth-order winding number of the Brunnian braid B 4 . The value of of this integral grows starting from zero when there is no entanglement and then eventually becomes one which is the desired result.
The fourth-order winding number given by Eq. (41) can be used to distinguish four unlinked strands from the entanglement of the Brunnian braid.

Cup product is a generalization of the linking number
Practical algorithm for finding a finite presentation of the fundamental group π 1 (X, x 0 ) of an arbitrary finite regular CWspace X which was illustrated in [21] and described in detail in [22]. From such a presentation, one can calcualate the cup product without need for any further significant computations since this product is essentially an invariant of π 1 (X, x 0 ) details in [23].  Figure 5 shows a pure permutahedral complex L representing a link with four components that represents the Brunnian link. To investigate the link we embed it into the interior of a contractible pure permutahedral complex R and form the complement M = R \L of the interior of L.
Groups, Algorithms, Programming GAP [24] is a system for computational Group Theory.
The following GAP session loads such a complex M from the file purecubicalcomplex.txt available at [25]. The pure permutahedral complex M. The session first constructs a smaller homotopy equivalent pure permutahedral complex XM M.
The space XM is constructed using a zig-zag defomration retract technique based on simple homotopy collapses which is described explicitly in [21]. The session then uses XM to compute the cohomology group H n (M, Z) and a presentation for G = π 1 M using the algorithm of [22].
gap> Read("purecubicalcomplex.txt");; gap> XM:=RegularCWComplex( ZigZagContractedComplex(M)); Regular CW-complex of dimension 3 gap> G:=FundamentalGroup(XM); <fp group of size infinity on the generators [f1,f2,f3,f4,f5,f6,f7]> The following continuation of the GAP session uses the method described in [23] to compute the cup product ∪ i α i where α i are free generators of H 1 (M, Z). It is well known that the cup product ∪ i α i can be interpreted in terms of the linking number Lk(K 1 , K 2 , K 3 , K 4 ) where K i are the four components in the Brunnian link (see for instance [26]). These GAP functions are provided by HAP [27] which is one of GAP packages.

Conclusions
It is important to look for high-order linking numbers to measure the entanglement of the field lines of the magnetic braid, and by which one can avoid the problem of the permutation. In this regard, we first found that the integral formula for a generalised Sato-Levine invariant is consistent in certain situations with Evans and Berger's formula for the fourth-order winding number. Also, we found that, in principle, one can go forward to derive analogous high-order winding numbers. For example we applied the fourth-order linking formula to Brunnian rings and verified that the linking number is 1. It can also be calculated by computing the cup product on the cohomology of a finite regular CW-space which is the complement R 3 \B 4 .