The wiki tells me that if you heat carbon at atmospheric pressure it eventually turns directly into a gas without being liquid first. At what pressure can you make liquid carbon? Has anyone actually ever studied liquid carbon? Surprisingly a web search did not answer these questions.
The carbon phase diagram shows that liquid carbon is only achievable at high pressure (~100 atm) and high temperature (~4500 K). This means that liquid carbon probably has very limited applications, and not many will be researching it. Funding is hard to come by when the applications of the research are not apparent.
“Surprisingly a web search did not answer these questions” — Well, a web search for liquid carbon turns a few results:
A 2005 study of liquid carbon by x-ray absorption spectroscopy (second Google search result) (the paper itself):
low-density liquid carbon contains predominantly twofold-coordinated chain structures (sp hybridization). As the density increased to that of solid forms, bond hybridization increased and threefold-coordinated (graphite-like) and fourfold-coordinated (diamond-like) bonds become more prevalent (indicating sp2 and sp3 hybridization, respectively). These observations are consistent with molecular dynamics calculations that rely on a tight-binding model of interatomic bonding. The fits also suggest that the bond length between carbon atoms in the liquid is significantly shorter than those in the solid, an observation also consistent with simulations.
An earlier study (1999) suggest two different liquid phases
So it's still a pretty open research question…