Predicting Financial Contagion and Crisis by Using Jones, Alexander Polynomial and Knot Theory


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.

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Vukovic, O. (2015) Predicting Financial Contagion and Crisis by Using Jones, Alexander Polynomial and Knot Theory. Journal of Applied Mathematics and Physics, 3, 1073-1079. doi: 10.4236/jamp.2015.39133.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Bhattacharya, R. and Majumdar, M. (2003) Random Dynamical Systems: A Review. Economic Theory, 23, 13-38.
[2] Crauel, H., Debussche, A. and Flandoli, F. (1997) Random Attractors. Journal of Dynamics and Differential Equations, 9, 307-341.
[3] Arnold, L. (2013) Random Dynamical Systems. Springer Science & Business Media, Berlin.
[4] Nechaev, S.K. (1996) Statistics of Knots and Entangled Random Walks. World Scientific, Singapore City.
[5] Kauffman, L.H. (2006) Formal Knot Theory. Courier Corporation, New York.
[6] Schenk-Hoppé, K.R. (1998) Random Attractors—General Properties, Existence and Applications to Stochastic Bifurcation Theory. Discrete and Continuous Dynamical Systems, 4, 99-130.
[7] Murasugi, K. (1987) Jones Polynomials and Classical Conjectures in Knot Theory. II. Mathematical Proceedings of the Cambridge Philosophical Society, 102, 317-318. Cambridge University Press, Cambridge.
[8] Labastida, J.M.F., Llatas, P.M. and Ramallo, A.V. (1991) Knot Operators in Chern-Simons Gauge Theory. Nuclear Physics B, 348, 651-692.
[9] Crowell, R.H. and Fox, R.H. (2012) Introduction to Knot Theory (Vol. 57). Springer Science & Business Media, Berlin.
[10] Webster, B. (2013) Knot Invariants and Higher Representation Theory. arXiv Preprint arXiv:1309.3796
[11] Cherednik, I. (2013) Jones Polynomials of Torus Knots via DAHA. International Mathematics Research Notices, 23, 5366-5425
[12] Katritch, V., Olson, W.K., Vologodskii, A., Dubochet, J. and Stasiak, A. (2000) Tightness of Random Knotting. Physical Review E, 61, 5545.

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