Conformal Geometry and Invariants of 3-strand Brownian Braids

摘要

We propose a simple geometrical construction of topological invariants of3-strand Brownian braids viewed as world lines of 3 particles performingindependent Brownian motions in the complex plane z. Our construction is basedon the properties of conformal maps of doubly-punctured plane z to theuniversal covering surface. The special attention is paid to the case ofindistinguishable particles. Our method of conformal maps allows us toinvestigate the statistical properties of the topological complexity of a bunchof 3-strand Brownian braids and to compute the expectation value of theirreducible braid length in the non-Abelian case.