Braided and Knotted Stocks in the Stock Market: Anticipating the flash crashes

A simple and elegant arrangement of stock components of a portfolio (market index-DJIA) in a recent paper [1], has led to the construction of crossing of stocks diagram. The crossing stocks method revealed hidden remarkable algebraic and geometrical aspects of stock market. The present paper continues to uncover new mathematical structures residing from crossings of stocks diagram by introducing topological properties stock market is endowed with. The crossings of stocks are categorized as overcrossings and undercrossings and interpreted as generators of braid that stocks form in the process of prices quotations in the market. Topological structure of the stock market is even richer if the closure of stocks braid is considered, such that it forms a knot. To distinguish the kind of knot that stock market forms, Alexander-Conway polynomial and the Jones polynomials are calculated for some knotted stocks. These invariants of knots are important for the future practical applications topological stock market might have. Such application may account of the relation between Jones polynomial and phase transition statistical models to provide a clear way to anticipate the transition of financial markets to the phase that leads to crisis. The resemblance between braided stocks and logic gates of topological quantum computers could quantum encode the stock market behavior.