How tangled is a tangle?

New measures of algebraic, geometric and topological complexity are introdu ced and tested to quantify morphological aspects of a generic tangle of fil aments. The tangle is produced by standard numerical simulation of superflu id helium turbulence, which we use as a benchmark for numerical investigati on of complex systems. We find that the measures used, based on crossing nu mber information, are good indicators of generic behaviour and detect accur ately a tangle's complexity. Direct measurements of kinetic helicity are fo und to be in agreement with the other complexity-based treasures, proving t hat helicity is also a good indicator of structural complexity. We find tha t complexity-based measure growth rates are consistently similar to one ano ther. The growth rate of kinetic helicity is found to be twice that of ener gy. (C) 2001 Elsevier Science B.V. All rights reserved.