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Recently a promising start-up (Ayasdi) has made headlines. They are a spin-off of the Applied and Computational Algebraic Topology group of Stanford University (ComTop). What they basically do is visualizing the topological structure of Big Data.

Now I know that this idea of applied (algebraic) topology crops up from time to time also in finance, see e.g. this blog post from Quantivity: Manifold learning.

There is even an R-package with which you can play around (the concept behind it is persistent homology which is a robust variant of the topological invariant): Phom.

My question
I am interested in applications of topological methods in finance, i.e. references (books, papers, articles) and software applications, that you can use to do some tests on your own.

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Stocks in the market can be twisted in braids and knots according to this paper http://arxiv.org/abs/1404.6637

Is a direct way to apply topology in finance.

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Interesting premise: a market metric based on Jones polynomial! I wonder if its behavior differs much from standard correlation metrics. Based on your username, the paper is your own -- it would be polite to mention that in your answer. – Brian B May 1 '14 at 13:07

You might want to check into Python Mapper, a Python module written by some of the founders of Ayasdi. It can be used to generate simplicial complexes which can be used to construct visualizations like those at Ayasdi.

I ended using Gephi, a network visualization software, to visualize the 1-simplexes generated from Mapper. As an example, the image below is the result of running mapper on a point cloud of data sampled from a torus.

enter image description here

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