Thysanotus is an Australian genus that has been listed as belonging to many different families including Liliaceae. You sometimes see it included in Anthericaceae which the Angiosperm Phylogeny Group II, 2003 suggests could optionally be included in the Agavaceae or Asparagaceae family. Another possibility is the Laxmanniaceae family where it is included with Lomandra. This genus has a common name of Fringe Lily and many of the species are climbing. Roots are fibrous, rhizomes, or tubers.
Thysanotus patersonii has tuberous roots and is native to Western Australia and Victoria. It is found in the wild growing through shrubs and has short-lived violet flowers with fringed inner segments borne on lateral stems. It is considered a short lived plant and this was true for me. I kept it alive for several years and it bloomed two of those seasons including spring 2004. It seems to be gone now. Photos by Mary Sue Ittner of my plant climbing and flowering on a wire fence.