A Method of Riemann–Hilbert Problem for Zhang’s Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure

A method of the Riemann–Hilbert problem is applied for Zhang’s conjecture 1 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in the zero external field and the solution to the Zhang’s conjecture 1 is constructed by use of the monoidal transform. At first, the knot structure of the ferromagnetic 3D Ising model in the zero external field is determined and the non-local behavior of the ferromagnetic 3D Ising model can be described by the non-trivial knot structure. A representation from the knot space to the Clifford algebra of exponential type is constructed, and the partition function of the ferromagnetic 3D Ising model in the zero external field can be obtained by this representation (Theorem I). After a realization of the knots on a Riemann surface of hyperelliptic type, the monodromy representation is realized from the representation. The Riemann–Hilbert problem is formulated and the solution is obtained (Theorem II). Finally, the monoidal transformation is introduced for the solution and the trivialization of the representation is constructed (Theorem III). By this, we can obtain the desired solution to the Zhang’s conjecture 1 (Main Theorem). The present work not only proves the Zhang’s conjecture 1, but also shows that the 3D Ising model is a good platform for studying in deep the mathematical structure of a physical many-body interacting spin system and the connections between algebra, topology, and geometry.


Introduction
The study on the Ising model has attracted intensive interest since the 1920s [1], which not only applies to interpret phase transitions and critical phenomena in different fields, but also provides fundamental understanding on interactions and dimensionality in nature. Onsager derived the exact solution of a two-dimensional (2D) Ising model in the zero external field, in which no non-trivial topological structures exist [2]. There had been no rigorous results on the solution of a three-dimensional (3D) Ising model in the zero external field, even for a simplest case with all interactions ferromagnetic. On the observation of the formula of the partition functions, the second author (ZDZ) made an observation that non-trivial topological structures exist in the ferromagnetic 3D Ising lattices in the zero external field for any positive inverse temperature [3,4]. This observation consists with the point view of Newell and Montroll [5], who pointed out that for the 3D Ising model in the zero external field, one encounters polygons with knots. Moreover, the second author conjectured that the non-trivial knot/link structures of the ferromagnetic 3D Ising model in the zero external field can be trivialized in higher dimensional space at the first beginning and the neglects of important non-locality factors during procedures. As pointed out in [30] that these estimates in [17] were obtained based on certain hypotheses (e.g., the existence of a sharp kink) and that if these hypotheses are not used, then the conformal bootstrap analysis appears to be consistent with the values η = 1/8 and ν = 2/3, obtained by Grouping of Feynman Diagrams, which are consistent with the Zhang's solutions obtained in [3]. Furthermore, Zhang's results agree with some experimental results, which are carefully performed with high accuracy (see in [31], for instance). After its publication [3], Zhang's conjectured solution has received supports from several groups, for instance, March and his co-workers [32][33][34][35][36], Ławrynowicz and some mathematicians [6,7,37,38], Kaupuzs and his colleagues [30,39,40], and others [41][42][43][44][45][46][47][48][49][50][51][52][53]. In [8], Zhang-Suzuki-March rigorously proved four Theorems, which verifies the correctness of the Zhang's conjectured solution [3]. The correct way to judge the correctness of an exact solution is to check whether there is anything wrong in the deriving process of proofs of Theorems, not to judge it by the approximant results. After the publication of Zhang-Suzuki-March's work [8], up to date, no further criticisms have been published. It was suggested that the approximation techniques can be utilized to obtain the non-locality part of the partition function (as well as the thermodynamic physical properties) by extracting the approximation values from the exact solution [54,55].
Although the Clifford algebra approach developed in [8] has already verifies the correctness of the exact solution based on the Zhang's two conjectures [3], it is still of great interest to prove the two conjectures from other mathematical aspects. In this way, the 3D Ising model and its exact solution would serve as a platform for investigating the connections between algebra, geometry, and topology, which are associated with the mathematical structure of the 3D Ising model. In this work, we will develop a method of Riemann-Hilbert problem for Zhang's conjecture 1, regarding the trivialization of the topological structure. The method of Riemann-Hilbert problem for Zhang's conjecture 2, regarding to the generation of topological phases, is in progress. In Section 2, the Hamiltonian and partition function of the ferromagnetic 3D Ising model in the zero external field are represented, and the Zhang's conjecture 1 is introduced with mathematical aspects. In Section 3, the Clifford algebra of the ferromagnetic 3D Ising model and its Knot/Clifford (K/C) algebra are constructed. In Section 4, the knot structure of the ferromagnetic 3D Ising model in the zero external field is investigated, and the partition function of the 3D Ising model can be generated by construction of the K/C-knots with the normal lattice knot γ, two types of basic knots (circles and braids), and their crossings. In Section 5, the knot/link structure is realized on a hyperelliptic Riemann surface, by the complex analysis. In Section 6, the method of the Riemann-Hilbert problem is applied for the representation. In Section 7, monoidal transforms are applied to give the desired trivialization of knot structures. In Section 8, the construction of solution to the Zhang's conjecture 1 is given, based on the procedures in the previous sections, which proves the Main Theorem, namely, that Zhang's conjecture 1 has been proven. In Section 9, the conclusion is given. The details for proofs of Theorems II and III are represented in Appendixes A and B, respectively.

3D Ising Model and Zhang's Conjectures
We recall some basic facts on the ferromagnetic 3D Ising model in the zero external field and represent its description by Clifford algebra.

Hamiltonian and partition function of 3D Ising model
We consider the orthorhombic lattice in the 3D Euclidean space [3,4]. Either up-spin or down-spin is located at each lattice point. The Hamiltonian of this statistical model is given as Here, we introduce variables ) [2][3][4]20,56]. We define the matrices j C and j s ' as follows: . Note that although only the nearest neighboring interaction between spins at each lattice point is considered in the Hamiltonian, the expressions of higher degree than quadratic in -matrices (i.e., non-local terms like the log of V3) appear in the partition function, which represent the existence of a long-range many-body entanglement between spins in the ferromagnetic 3D Ising model, because of the nature of three dimensions.

Zhang's conjecture 1
Here, we discuss the main problem on a ferromagnetic 3D Ising model in the zero external field and state the Zhang's conjecture 1. Although the ferromagnetic 3D Ising model we studied has the nearest neighboring interaction only, the nature of three dimensions results in two different behaviors for the interactions: (i) j C or represents the nearest interaction along the first or second dimension, and (ii) represents the nearest interaction along the third dimension, which consists of the non-local behavior, namely, a kind of long-range many-body entanglement. The interaction j C or can be described by use of the Lie group/algebra (so-called Gaussian type). The main concern is , which leads to our main problem of trivialization for the non-trivial knot structure. Zhang has discussed this problem in terms of topology [4], namely, the knot theory [57,58]. Topologically, there are two choices for smoothing a given crossing (), and thus there are 2 N states of a diagram with N crossings [57,58]. The bracket state summation is an analog of a partition function in discrete statistical mechanics, which can be used to express the partition function for the Potts model for appropriate choices of commuting algebraic variables [57,58]. This means that not only the local spin alignments but also the crossings of knots contribute to the partition function Z of the 3D Ising model in the zero external field. The contribution to the partition function Z by knots also reflects the entropy cost of tying knots, as the partition function Z is related with the free energy F, the internal energy U, and the entropy S.
In this paper, we introduce the knot structure which is called "Knot/Clifford (K/C) knot" and associate the knot for the partition function. We can have the representation from the K/C knot to Clifford algebra, and describe the topological structure of the partition function of the ferromagnetic 3D Ising model in the zero external field, which might give the answer of the problem by making the trivialization of the given (non-trivial) knot. The trivialized knots constitute with the knot of trivial type only.
Consider the 3D lattice Z3 and take a knot γ which is constructed by the horizontal line and vertical joint line with vertex {Pj}, which is denoted by γ = {Pj}, as illustrated below in Figure 1, for example. Generators of the transfer matrices V1, V2, and V3 are associated to some points in γ, which are denoted by X = {Vi (j) }. Then, we have a data (γ, X) which is called Knot/Clifford (K/C) data,  Z . The Zhang's conjecture 1 can be stated in the following manner: For a given  Z , can we make a trivialization Z ?
More exactly, we can state the Zhang's conjecture 1 as: Can we find a four-dimensional manifold 4 M , and a K/C data where ~ is usually a nontrivial knot (however, it can also be a trivial one for simple cases) and  is a member of generators satisfying the following condition.
We can find a trivialization mapping,

Steps to prove Zhang's conjecture 1 on trivialization
In this paper, we shall give a positive answer to the Zhang's conjecture 1 in the following steps: (1) The Clifford algebra ) ( 3D I Cl is extended to the K/C algebra which has the original Clifford algebra and its conjugate algebra ) ( 3D I Cl as subalgebras (Section 3).
(2)  Z is extended to the K/C algebra which is denoted by . Therefore, we have a knot carrying the elements in K/C algebra for the partition function (Section 4).
(3) After the realization of the knot on a Riemann surface (Section 5), we formulate the Riemann-Hilbert problem for the representation and obtain the solution (Section 6). (4) Applying the monoidal transformation to the solution in (3), we construct the desired trivialization in K/C algebra (Section 7).

Clifford Algebra of the Ferromagnetic 3D Ising Model and Its K/C Algebra
In this section, we state the generation of Clifford algebra of the ferromagnetic 3D Ising model in the zero external field by the basic construction and then give its extension which is called a K/C algebra.
Repeating the basic construction successively, we can obtain general Clifford algebra: Following the notation due to Onsager-Kaufman-Zhang [2][3][4]8,56], we choose the no- Therefore, we can introduce the following generators of Clifford algebra of the 3D Ising model:

K/C algebra associated to the knot structure
We introduce the K/C algebra which is an extension of Clifford algebra of the ferromagnetic 3D Ising model. We put the generators of the Clifford algebra and its conjugate elements: In order to obtain the commutation relations   j i  , , we introduce the following algebra with the product table which is called knot-algebra (see Table 1).
The product is defined by b a  . For example, We can obtain the following proposition.
(2) We have Here, we have calculated the product including ' ' , ' s s as the usual matrix calculus. It implies that the first element determines the sequence to be -sequence or  -sequence.
In the following we call -sequence positive sequence,  -sequence negative sequence, respectively.
The K/C algebra has the following commutation relation: It is seen that the K/C algebra has subalgebras which are real Clifford algebra and its conjugate algebra, and they are conjugate each other.

Knot Structure of the Ferromagnetic 3D Ising Model
In this section, we introduce a special class of knots, which are called knots of the ferromagnetic 3D Ising model in the zero external field, by use of the product structure of -factors. Then, the representation from the K/C algebra of 3D Ising type to the knots of 3D Ising type can be found.
The topological structures of the ferromagnetic 3D Ising model in the zero external field are constructed by two parts of contributions. First, there exist normal (either trivial or nontrivial) knots γ (or links), which are generated by lattice points. Second, the Pauli matrices themselves may be treated as crossings [57,58], which also contribute the fine structure of the topology. The latter case is the main concern of this paper, as the nonlinear terms in the partition function contribute the non-trivial topological structure of the ferromagnetic 3D Ising model in the zero external field.

Knots with Clifford algebra data
In this paper, when we say a knot, it has always Clifford algebra data. We choose a knot γ which is generated by lattice points {P1, …, Pn}. Namely, γ = P1, …, Pn. Choosing , and making their conjugates

Association of circle/interval to -factors
We choose an element ) ( 3D I Cl   and associate a circle (or interval) to .
and associate a dotted circle (or dotted interval):

Generation of knots of ferromagnetic 3D Ising model for partition functions
We treat the knot structure of the partition function of the ferromagnetic 3D Ising model in the zero external field as follows. There exist two types of basic forms: the ele- The detailed construction of this knot structure from Pauli matrices in the transfer matrices V1 and V2 are omitted for simplicity. The elements ) (with j = 1,2,…,nl; k = 2n) together with their conjugate elements are called basic form of type II (see Figure 3), which can be represented as a braid with many crosses (k = 2n). Indeed, each factor of iΓ Γ in the internal factors of the transfer matrices V3 is equal to the Pauli matrix [13], contributing a crossing topologically [57,58]. Therefore, each term of exponential elements in the transfer matrices V1 and V2 contributes a circle to the knot structure, while each term of exponential elements in V3 contributes a braid. There is also a type which is defined by the product of exponential elements generated by the basic types above. The circles in V1 and V2 can be adjoined together with the lattice points of the normal knots γ, while the braids in V3 can be connected as the product type of knots. As an example, the following figure ( Figure 4) just shows three of the braids in V3 connecting to the lattice points of the knot γ, while it does not show the circles of V1 and V2 for simplicity. Note that the knot structure of the ferromagnetic 3D Ising model in the zero external field is much more complicated than what we illustrated in the above figure. The other end of these braids should be connected to other lattice points in accordance with the expression for the partition function. Adjoining the braids for different j in V3 forms new crosses, making the topological structure much more complicated. We can introduce a concept of confluency of knots and discuss the generation of K/C knots by use of successive confluence operations, to obtain the K/C knots of the generators Vi (i = 1,2,3) of the partition functions of the ferromagnetic 3D Ising model in the zero external field. Nevertheless, the process for trivialization of the above topological structure can be employed directly to trivialize the more complicated one, as the concept, the principle, the role, and the process are kept the same.
By these observations, we see the following theorem.
Theorem I. The partition function of the ferromagnetic 3D Ising model in the zero external field can be generated by construction of the K/C-knots with the normal lattice knot γ, two types of basic knots (circles and braids), and their crossings. The transfer matrices V1 and V2 contribute the trivial parts to the topological structure of the ferromagnetic 3D Ising model in the zero external field, while the transfer matrices V3 contribute the non-trivial knots to the system.

Realization of Knots on a Riemann Surface
In this section, the knot/link structure is realized on a hyperelliptic Riemann surface. This is the first step of the realization of knot/link structure by the complex analysis.

Realization of knots
We choose a knot ξ, which is expressed as Assume that ξ is a positive sequence, and that The intersection points are denoted as a1, a2, ……, an. We prepare two copies of complex projective space P 1 . A 2-covering Riemann surface Mg is made by use of cut-segments: In the case where n is even (n = 2m), one has Figure 5. In the case where n is odd (n = 2m + 1), one has Figure 6. In a well-known manner, make a Riemann surface. We demonstrate in Figure 7 the construction of the Riemann surface z : in this case, we make the cut along (0∞) and glue in the figure. As for a general case, the construction is performed in a completely analogues manner.

Realization of Knots on a Four-dimensional Manifold
In order to make clear the role of monoidal transform, we realize the base Riemann surface Mg in P  C as a covering space over P 1 (Figure 8). Then, we assume that the knot has a singularity of normal crossing. C P M g   Figure 8. The base Riemann surface Mg in P  C as a covering space over P 1 in which the knot has a singularity of normal crossing.    (4) Repeating this process, we obtain a closed curve which is located very near to the original knot curve.

Remark 1.
Here, we have to pay attention to check whether entanglement appears or not. We will not be concerned with this problem. This is because when the entanglement points appear, we extend the knot and associate the trivial matrix and solve our problem.
Example. We choose a knot sequence in Figure 11: a a a a a a a a  . Figure 11. A knot with crossings.
We prepare two sheets of P 1 and make cuts between  Figure 12, we can obtain the realization of the knot in Figure 11 on the Riemann surface. We can prove the following Proposition.

Proposition. From the representation
such that the following commutative relations hold:

Method of Riemann-Hilbert Problem for 3D Ising Model
In this section, the method of the Riemann-Hilbert problem is applied for our representation. At first, let us recall the Riemann-Hilbert problem.

Riemann-Hilbert problem
As for a function f on a compact Riemann surface Rg which has a regular singularity at a1, a2, …, aM. The function has the following form: We have the monodromy representation, is the fundamental group. The Riemann-Hilbert problem asks the converse problem [59,60]: "when a monodromy representation is given, can we find a multi-valued  [59,60].

Remark 2.
The solution may have additional singularities which are rational singularities with trivial local monodromies, which is called "accessory singularity".

Riemann-Hilbert problem for the ferromagnetic 3D Ising model
The Riemann-Hilbert problem can be formulated for our representation and it can be solved. The following theorem can be proven:

Theorem II. The Riemann-Hilbert problem is applied for the representation on the ferromagnetic 3D Ising model in the zero external field.
(1) For a given generator Vi (i = 1,2,3) of the partition function of the ferromagnetic 3D Ising model in the zero external field, we find a knot i  , which is given in Section 4. After making the realization of knots on a Riemann surface, we can find the following representation:

Remark 3. We can treat knot points which arise from the entanglement by use of singularities. In fact, when a given knot is realized on the Riemann surface, we have entanglements. Then, we consider an extended knot which includes entangle knot points and associate the trivial identity matrix
as i  or i  , the same process can be performed.
Note that although Riemann-Hilbert problem, aiming to keep under control nonlinearity, is a technique designed for the differentiable continuum rather than a lattice, in the thermodynamic limit the 3D Ising model on a manifold can be treated as a differentiable continuum, including its partition function, the transfer matrices and the K/C-knots.

Construction of Trivialization by Monoidal Transforms
In this section a concept of monoidal transform is introduced. It will be shown that successive applications of the transforms give the desired trivialization.

Direction separation-basic idea
There is a curve which has a normal crossing at 0. By use of the following mapping, Q: We can separate the direction at 0 and get the desired trivialization ( Figure 13):

Take a point Po, we choose a local coordinate
is made to satisfying the following condition: There exists a mapping (see Figure 14) ) ( ) ( : Figure 14. A mapping for monoidal transform.

Complex line bundle of monoidal transformation
In consideration of the following complex line bundle over P: F    P, the tuber neighborhood 0 ( ) U P  satisfies the conditions (i) and (ii). The construction of π: F → P is given as follows: Let {V0,V∞} be the standard local coordinate system of  1 : P We have a complex line bundle:

Construction of monoidal transform
We have the desired mapping

Remark 4. We make stress on the following fact: Even if M = C 2 (= R 4 ), the manifold M is not
Euclidean and non-trivial topology appears.

Basic Notations on Trivialization
For the definition of the concept of trivialization, we begin with some basic notations: (1) Trivial elements of exponential type A direct sum of elements of exponential type of basic 1-form: is called trivial element. The following mapping: is called generation mapping.
(2) K/C mapping The set of K/C elements on a manifold M is denoted by K/C(M). Then, we have Let M and M' be two manifolds, and let K/C(M) and K/C(M') be the K/C elements.

Basic Idea on Trivialization
Before giving to the statement of Theorem, we demonstrate the basic idea on the trivialization by monoidal transforms in the case of basic type. We consider the following configuration: As have remarked, we may assume that the Riemann surface is realized in P  C and the knot has singularities of normal crossing ( Figure 15): We consider the solution of the Riemann-Hilbert problem: and next make the monoidal transform ( Figure 16) at the intersection point c1 in Figure 15:  Therefore, we can make the following identification ( Figure 18): Proof of Theorem III is given in Appendix B. Finally, the trivialization mapping can be constructed in the explicit manner for the transfer matrices V = V1V2V3. The process can be performed in a completely analogous manner, but it is much more complicated, as there are many products of knots.

Remark 5.
Decomposing the trivialization into Clifford algebra and its conjugate algebra, we have Z and Z . Then it is seen that Z is the desired trivialization: As mentioned in the previous sections, there exist normal knots γ (or links), which are generated by lattice points. Furthermore, the Pauli matrices themselves may be treated as crossings, which contribute the fine structure of the topology. The topological structure of the ferromagnetic 3D Ising model is much more complicated than the examples above. Nevertheless, the trivialization procedure is in an analogous manner, and it can be generalized to be appropriate for any knots (and links) which fit with the partition function (together with normal knots/links γ in lattice points) of the 3D Ising lattice. The monoidal transform utilized here for trivialization of knots consist with the Clifford algebra approach developed in [3,4,8], in which the topology theory [57,58] is employed to take into account of the contribution of non-trivial knots to physical properties of the 3D Ising model, and the generalized Yang-Baxter equations (i.e., the so-called tetrahedron equations) [76][77][78][79][80], corresponding to a Reidemeister move of type III, are used to guarantee the integrability of the system [4]. However, the generalized Yang-Baxter equations [76][77][78][79][80] involve not only the Reidemeister moves, but also the disconnection and fusion of crossings (which can be mapped to the creation and annihilation of spins or particles). Furthermore, the tetrahedron equations are highly overdetermined and constraints must be imposed on the variables of each local weight to allow for a solution [79]. These make the difficulties of explicitly solving the 3D Ising model. In [8], we introduced a local transformation as a topological transformation to deal with the crossings in the 3D Ising model, in order to satisfy the requirements of the tetrahedron equations and to take into account the contribution of these crossings to the physical properties. Meanwhile, we utilized the Largest Eigenvalue Principle that only the largest eigenvalue contributes dominantly to the partition function of the 3D Ising model in the thermodynamic limit. With this constraint, we pick up the desired solution among all 2 nl possible solutions in 2 nl sub-spaces produced by the direct product of all the sub-transfer matrices. The combination of the local transformation and the Largest Eigenvalue Principle solves the problems of overdetermined tetrahedron equations. The exact solution we found is , which is a star-triangle relationship and a solution of the tetrahedron equations and also the Yang-Baxter equations [4,8].

Construction of Solution to the Zhang's Conjecture 1
With the preliminaries in the previous sections, we can construct the trivialization for the Zhang's conjecture 1 in the following steps: (1) We assume that a knot γ is given on the lattice: γ = {Pi}, where Pi is on the lattice points.
(2) We take a partition function which is defined by the transfer matrices V1, V2, and V3.
The members are denoted by V (i) (i = 1,2,…, M). Summarizing the procedures of this paper, we have proved the following Theorem.
Main Theorem: The Zhang's conjecture 1 proposed in [3] can be solved. Namely, for a given K/C knot which is given by the partition function of the ferromagnetic 3D Ising model in the zero external field, we can make the trivialization by use of monoidal transform trivialization.

Conclusions
In conclusion, we have developed a method of Riemann-Hilbert problem for Zhang's conjecture 1 proposed in [3], regarding to the trivialization of topological structure, for the ferromagnetic 3D Ising model in the zero external field. Three claims are formulated in form of Theorems: (I) The partition function Z of the ferromagnetic 3D Ising model in the zero external field can be generated not only by spin alignments, but also by knots. In between these knots one has components which have a complicated topological structure that is contributed by nonlinear terms in the transfer matrices V3. (II) In order to relax this complicated structure, realizations of knots are produced on a four-dimensional Riemann manifold, which are formulated in the Riemann-Hilbert problem for the representation. (III) The monoidal transformations are applied at the knot intersection (singular) points, eliminating these from knots, thus producing the trivialization of the knots. The immediate consequence of these claims is the main theorem that the Zhang's conjecture 1 proposed in [3] has been proved. The explicit expression for the resulting partition function Z have not been provided directly by the present procedure of the Riemann-Hilbert problem and the monoidal transformations. The partition function Z in a 4-fold integral form was presented in Equation (49) in [3] (see also Equation (24) in [4]) based on the Zhang's two conjectures, which was proven to be correct in [8] by a Clifford algebra approach. The thermodynamical properties (including the free energy, the specific heat, the spontaneous magnetization, the spin correlation, the susceptibility, as well as the critical exponents) of the ferromagnetic 3D Ising model in the zero external field are derived explicitly in [3]. However, attention should be paid on using the present procedure to demonstrate rigorous formulation for the non-trivial knots' components of the partition function Z. A subsequent paper, on the method of Riemann-Hilbert problem for Zhang's conjecture 2, regarding to the generation of topological phases, will be published soon [81]. Furthermore, because the exact solution for the antiferromagnetic 3D Ising model with all the negative interactions but without frustration in the zero external field has the same formula as the exact solution obtained in [3,4,8], the results in [3,4,8] for the ferromagnetic 3D Ising model in the zero external field are suitable for the antiferromagnetic 3D Ising model without frustration.
It should be emphasized that the procedures developed in [8] and this work, to deal with the non-trivial topological structures in the ferromagnetic 3D Ising lattices in the zero external field, and the exact solution obtained in [3], are valid for any positive inverse temperature β = 1/(kBT) > 0 (including the area of phase transitions, βc), where the nontrivial topological structures exist. It is clear now that the non-trivial topological structures (knots) contribute additional terms to the partition function and the physical properties (such as the free energy, the specific heat, the spontaneous magnetization, the spin correlation, the susceptibility, the critical exponents). Any approaches based on only local environments, such as conventional low-temperature expansions, conventional high-temperature expansions, Monte Carlo simulations, Renormalization Group, etc., are not exact, because these approximative approaches miss the contributions of knots. The conventional high-temperature expansions work only at/near infinite temperature β = 0, where only the trivial topological structure exists. Our procedures in [3,4,8] and the present work indicate that it is necessary to introduce an additional dimension (the time) to trivialize the knots and take into account their contributions, to achieve the exact solution of the 3D Ising models. The temperature-time duality in the 3D Ising model can be seen by inspecting the resemblance between the density operator in quantum statistical mechanics and the evolution operator in quantum field theory, with the mapping β = (kBT) −1  it = τ [82][83][84]. In [84], Zhang and March pointed out that besides the Wick rotation, which represents the temperature as the imaginary time, we have to introduce also the time for the time average and for untying the knots (see also in [3,4,8] and this work). Therefore, one has to deal with the topological quantum field theory within a (3 + 2) or (4 + 1)-dimensional framework [85]. With Wick representation, the ε-expansions [86][87][88] start from four dimensions and do not account the non-trivial topological contributions, which are still in the approximative level. It can be improved by accounting the contributions of the nontrivial topological structure in the (3 + 2) or (4 + 1)-dimensional framework. Finally, we would like to notice that our work illustrates that the 3D Ising spin system can serve as a platform for describing a sensible interplay in between the physical properties of interacting many-body systems, algebra, topology, and geometry.

Data Availability Statement:
The data is available on reasonable request from the corresponding author.

Conflicts of Interest:
The authors declare no conflicts of interest.

Appendix A. Proof of Theorem II
Appendix A. 1

. Construction of the Representation (Proof of (1) in Theorem II)
We give the construction  which is given in (1) of Theorem II and make the twisted knot from the standard knot. Then the Riemann-Hilbert problem can be formulated. We put (1) The case of basic type We make the knot of twisted type from the knot of standard type ( Figure A1): Figure A1. Making the knot of twisted type from the knot of standard type.
Next construct a Riemann surface. We prepare two branch points  The over (under) going path from P (respective Q) through P1 to Q (respective P) is denoted by (2) The case of simple/multi-type We choose the knot of multi-type with the following orientation of twist type: We consider the case m = 4 (see Figure A3). The over (under) going path from P (respective Q) to Q (respective P) is denoted by We choose a realization on a Riemann surface in the following manner: (i) Choose a Riemann surface ( Figure A4) Figure A4. A realization on a Riemann surface.
(ii) Make a realization of γ by use of association of twisted type ( Figure A5). The approach above can be applied for the general case of m.

Appendix A.2. Construction of the Solutions (Proof of (2) in Theorem II)
The solution of Riemann-Hilbert problem can be constructed. We give the solutions in the cases of basic types I and II, separately. We have to pay attention to the method of the construction. We construct the solutions both for positive elements and negative elements at the same time. Therefore, the solutions are considered in the universal K/C algebra.
(1) The solution for the basic form of type I The solution is given as follows: Putting  Then, following the procedure above, we can formulate the Riemann-Hilbert problem and find the desired solution. The discussion is completely identical, and it may be omitted.

Appendix B. Proof of Theorem III
We begin with recalling the classification of the knots in the transfer matrices V1, V2, and V3. From the results in Section 4, there are knots of the following types: basic types I and II, product type (and adjoint type). In the following, we construct the trivializations for these basic types and combining the results, the desired trivialization is obtained for the product type.
The configuration of type I can be trivialized by the scheme in the basic idea. We have the following trivialization ( Figure A6): Figure A6. The trivialization of the configuration of type I.
We construct the trivialization in the case of the configuration type II and m = 3 as the following process ( Figure A7). The desired trivialization is given by We treat the product type by the choice of an example: ) ) ... The trivialization can be made for each block (I), (II), or (III), separately, and joint each construction. The joint is given as follows: At each point Pi, we have no information on knot structures. At each point, we associate the trivial matrix which is denoted by ) ( i P  and ) ( i P  , make extensions, for example, the extension is given at Pi in the following manner and other cases are treated in a completely analogous manner ( Figure A9).
Therefore, the representation does not change, i.e., is invariant. The monoidal transform becomes as follows (see Figure A10): Here, the dotted circle implies the S 2 (  P 1 ) which does not contribute to the partition functions. Thus, we may omit it.
Performing the process to P2, P3, the following diagram is obtained by monoidal transforms (Figure 11): Figure A11. The monoidal transform for a configuration.