How tangled is a tangle?

New measures of algebraic, geometric and topological complexity are introduced and tested to quantify morphological aspects of a generic tangle of filaments. The tangle is produced by standard numerical simulation of superfluid helium turbulence, which we use as a benchmark for numerical investigation of complex systems. We find that the measures used, based on crossing number information, are good indicators of generic behaviour and detect accurately a tangle's complexity. Direct measurements of kinetic helicity are found to be in agreement with the other complexity-based treasures, proving that helicity is also a good indicator of structural complexity. We find that complexity-based measure growth rates are consistently similar to one another. The growth rate of kinetic helicity is found to be twice that of energy. (C) 2001 Elsevier Science B.V. All rights reserved.