ICMCS Seminar -Topological Measures of Entanglement in Systems of Curves

Many filamentary systems, including polymers and proteins, may derive many of their characteristic properties and function from the complex ways in which the constituent filaments (curves) wind around themselves and each other. The theory of mathematical knots provides powerful tools for studying entanglement in closed curves through topological invariants, but many systems found in nature are better modeled as open curves, which are topologically trivial in the classical sense. In this talk, I will briefly introduce knot theory and the notion of topological invariants before describing recent developments that extend these ideas to open curves. These closure-independent measures of entanglement generalize classical invariants, including the Jones polynomial. These measures change continuously with the geometry of the curves and recover classical knot and link invariants when the open endpoints tend to converge, providing a useful way to quantify and track changes in entanglement as filamentary systems evolve. Finally, I will discuss how machine learning can be used to predict topological complexity directly from geometry. Because the exact computation of topological invariants can be prohibitively expensive for large and complex systems, machine learning provides a scalable approach for rapidly characterizing entanglement, while also offering the potential to discover new relationships between geometry and topology.