A trivial tangle in a genus-g handlebody is determined, up to isotopy rel boundary, by the surjection it induces from the surface group onto the free group of rank g; such a surjection is called a bounding homomorphism, and Stallings folding decides in polynomial time whether a given homomorphism is one and reconstructs the tangle. Assembling two, three or more such homomorphisms along a common surface group turns Heegaard splittings of 3-manifolds, bridge splittings of links, trisections of 4-manifolds and bridge trisections of knotted surfaces into purely group-theoretic data, giving bijective correspondences between the topological objects and equivalence classes of tuples of bounding homomorphisms. The payoff is that questions about smooth 4-dimensional objects - whether a surface is unknotted, whether two knotted surfaces are equivalent, whether a smooth structure is standard - become finite algebraic questions about free and surface groups, at the cost of moving the difficulty into decision problems that are themselves undecidable in general.
Low-dimensional topology studies manifolds of dimension three and four. A Heegaard splitting cuts a closed oriented 3-manifold into two handlebodies glued along a common genus-g surface; a trisection…