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Topological methods are rapidly developing and are becoming more used in physics, biology and chemistry. One area of topology has showed its immense potential in explaining potential financial contagion and financial crisis in financial markets. The aforementioned method is knot theory. The movement of stock price has been marked and braids and knots have been noted. By analysing the knots and braids using Jones polynomial, it is tried to find if there exists an untrivial knot equal to unknot? After thorough analysis, possible financial contagion and financial crisis prediction are analysed by using instruments of knot theory pertaining in that sense to Jones, Laurent and Alexander polynomial. It is proved that it is possible to predict financial disruptions by observing possible knots in the graphs and finding appropriate polynomials. In order to analyse knot formation, the following approach is used: “Knot formation in three-dimensional space is considered and the equations about knot forming and its disentangling are considered”. After having defined the equations in three-dimensional space, the definition of Brownian bridge concerning formation of knots in three-dimensional space is defined. Using analogy method, the notion of Brownian bridge is translated into 2-dimensional space and the foundations for the application of knot theory in 2-dimensional space have been set up. At the same time, the aforementioned approach is innovative and it could be used in accordance with stochastic analysis and quantum finance.

In this paper, random dynamical systems are considered. It is assumed that financial time series exhibit fractional Brownian motion and knot theory is used in order to analyse the formation of knots in financial time series. The foundations are set up to further the analysis of the financial time series using quantum physics, knot theory and topology. Firstly, we will define mathematically random system, afterwards Wiener process via stochastic differential equation is defined and ordinary Brownian motion is at the same time defined. As ordinary Brownian motion is a subclass of fractional Brownian motion, fractional Brownian motion is explained. In the theory section, one question was posed and it stated: “What would happen if the time series pertaining in that sense to major financial indices follow fractional Brownian motion and are forming knots? Can the financial crisis be predicted by observing knot formation?” Afterwards, we proceed to analysis and formation of knots in three- dimensional space, we present the equations that represent the formation of 3 dimensional knots by using for- mulas from quantum physics and afterwards and we make the Brownian bridge hypothesis in three-dimensional space and translate it to two-dimensional space. With the aforementioned, thesis for analysis of two-dimensional knot set-up is defined and hopeful. The following papers of the author will approach the further analysis and development of equations in the formation of knots in two-dimensional space.

Random dynamical system [

If we want to implement a solution to stochastic differential equation, firstly some definitions should be set-up:

Let

exists for all positive time and some (small) interval of negative time dependent upon

In this context, the Wiener process is the coordinate process.

Now define a flow map or (solution operator) [

Then,

Components of a random dynamical system [

Let

Assume the following:

1)

2) For all,

That is

Now let

For all

For all

In case, we are considering a random dynamical system driven by Wiener process

This says that

The fBm is an extension of the classical Brownian motion that allows its disjoint increments to be correlated. Fractional Brownian motion is not Markovian and this becomes a strong difficulty to study and put the model into practice.

A centered Gaussian process

Usually it is assumed that

If

If

The main question that is posed what would happen if the time series pertaining in that sense to major financial indices follow fractional Brownian motion and are forming knots? Can the financial crisis be predicted by observing knot formation?

Conjecture 1: (Frisch-Wasserman Delbruck (FWD) Conjecture) [

The probability to find a closed N-step random walk in

where

In three-dimensional space, the following expression is found for [

Introducing the time,

If phase trajectories can be mutually transformed by means of continuous deformations, then the summation should be extended to all available paths in the system but if the phase space consists of different topological domains, then the summation in the above equation refers to the paths from the exclusively defined class and knot entropy problem arises [

The 2D version of the Edward’s model is formulated as follows. Take a plane with an excluded origin, producing the topological constraint for the random walk of length L with the initial and final points

In the said model, the state C is fully characterised by number of turns of the path around the origin. The corresponding abelian topological invariant is known as Gauss linking number and when represented in the contour integral form, reads [

where

And

Using the Fourier transform of the

We introduce the entropic force:

Which acts on the closed chain

Distribution function

Therefore according to D. S. Khandekar and F. W. Wiegel again represented the distribution function in terms of the path integral (Equation (20) with the replacement:

where the

The final distribution function reads to [

where

And

There is no principal difference between the problems of random walk statistics in the presence of a single topological obstacle or with a fixed algebraic area-both of them have the “abelian” nature.

The principal difficulty connected with application of the Gauss invariant is due to its incompleteness.

Any closed path on

into account

Application of Gauss invariant is due to its incompleteness. It has been recognized that the Alexander polynomials [

The probability to find a randomly generated knot in a specific topological state. Take an arbitrary graph and assume the following theorem: Two knots embedded in

Two knots are called regular isotopic if they are isotopic with respect to two last Reidemeister moves(II and III); meanwhile, if they are isotopic with respect to all moves, they are called ambient isotopic. As it can be seen from

Now, after the Reidemeister theorem has been formulated, it is possible to describe the construction of polynomial “bracket” invariant in the way proposed by L. H. Kauffman [

For the knot diagram with N vertices there are

Completed by intial condition:

where O denotes the separation of trivial loop.

Actually, the knot structure is formed during the random closure of the path and cannot be changed without the path rupture.

Knot complexity, the power of some algebraic invariant

One and the same value of

Take a set of knots obtained by closure of

Actually, the conditional probability distribution

Recall that the distribution function

With the diffusion coefficient

The diffusion equation for the scalar density

where

And

3D space:

The Brownian brigde condition for random walks in space of constant negative curvature makes the space “effectively flat” turning the corresponding limit probability distribution for random walks to the ordinary central limit distribution.

This question is valid in Euclidean space. If we translate it into two-dimensional space, the following result is obtained:

The Brownian bridge condition for random walks in 2-dimensional space makes the corresponding limit probability distribution for random walks to the ordinary central limit distribution.

The above mentioned equations have set up the foundations of applying knot theory to financial time series analysis. Firstly, the set-up for forming knots in three-dimensional space was performed using quantum physics tools and afterwards the set-up was translated into the 2-dimensional space. Brownian bridge was defined in

I would like to thank my family for the support, especially my father, mother, sister and aunt.

OgnjenVukovic, (2015) Predicting Financial Contagion and Crisis by Using Jones, Alexander Polynomial and Knot Theory. Journal of Applied Mathematics and Physics,03,1073-1079. doi: 10.4236/jamp.2015.39133