Algebraic representation formulas for null curves in Sl(2,ℂ)
We study curves in Sl(2,ℂ) whose tangent vectors have vanishing length with respect to the biinvariant conformal metric induced by the Killing form, so-called null curves. We establish differential invariants of them that resemble infinitesimal arc length, curvature and torsion of ordinary curves in Euclidean 3-space. We discuss various differential-algebraic representation formulas for null curves. One of them, a modification of the Bianchi-Small formula, gives an Sl(2,ℂ)-equivariant bijection between pairs of meromorphic functions and null curves. The inverse of this formula is also differential-algebraic. The other one is based on an integral formula deduced from that of R. Bryant, using certain natural differential operators on Riemannian surfaces that we introduced in  for differential-algebraic representation formulas of curves in ℂ³. We demonstrate some commands of a Mathematica package that resulted from our investigations, containing algebraic and graphical utilities to handle null curves, their invariants, representation formulas and associated surfaces of constant mean curvature 1 in ℍ³, taking into consideration several models of ℍ³.