Parity and Patterns in Low-dimensional Topology

Algebraic and topological objects are usually encoded by diagrams and moves (words and relations, etc). Diagrams (words) consist of nodes (crossings, letters). The parity theory initiated in 2009 by the second named author (V.O.Manturov) argues that if there is a smart way to distinguish between even and odd nodes (crossings, letters) in a way consistent with moves then this allows one to construct functorial mappings between objects of the theory, construct various powerful invariants, reduce problems about objects (say, knots) to problems about their diagrams, refine many existing invariants. Over the last six years, parity theory has experienced a rapid growth;investigations were undertaken by dozens of scientists worldwide. Various problems in low-dimensional topology were solved by using parity.