Tangles offer a precise way to identify structure in imprecise data.

By grouping qualities that often occur together, they can reveal not only clusters of things but also types of their qualities: types of political views, of texts, of health conditions, or of proteins. Tangles offer a new, structural, approach to artificial intelligence that can help us understand, classify, and predict complex phenomena.

Tangles can be used in conjunction with existing methods based on neural networks, or offer an alternative, when more control or accountability is desired.

The mathematical theory of tangles has its origin in the ground-breaking work of Neil Robertson and Paul Seymour on graph minors in the late 20th century. Its potential for applications was realised only more recently, following the axiomatization of tangles as abstract connectivity structures independent of graphs.

Such potential applications, which can be either

• generic in data science and machine learning; or
• context-specific in the natural sciences, the social sciences, or economics

are explored here for the first time. There are two formats, closely interlinked:

• the tangle applications book (R.Diestel, CUP 2024);
• tangle software that implements the algorithms described in the book, and more.