Telekinesis, Eyebite, and Bigby's Hand are some of the few spells which persistently affect one creature. As such, they appear to fit the requirements of Twinned spell, which are…
Twinned Spell:
When you Cast a Spell that Targets only one creature and doesn’t have a range of self, you can spend a number of sorcery points equal to the spell’s level to target a second creature in range with the same spell (1 sorcery point if the spell is a cantrip).To be eligible, a spell must be incapable of targeting more than one creature at the spell’s current level. For example, Magic Missile and Scorching Ray aren’t eligible, but Ray of Frost is.
Bigby's Hand, Eyebite, and Telekinesis are all incapable of targeting more than one creature at a time at any level. This leads to…Difficulties…interpreting then. Can you even twin them? If you can twin them, how do you target them on future turns? Can you cause different effects to different targets on each turn, per Eyebite or Bigby's?
I included Bigby's because of multiclassing.
Best Answer
You cannot twin any of these spells
Eyebite has a range of self so it is explicitly disallowed when twinning as per the Twinned Spell description:
Bigby's Hand and Telekinesis do not target a creature, but rather each create an effect, which can then target a creature. This would seem to disallow either of them from being twinned.
In addition, they both could affect multiple creatures throughout the length of the spell, which should disqualify them as well. As per the Twinned Spell description,
This does not say "incapable of affecting more than one creature at a time." Additionally, the extra rules that would be required in order to dictate how they work on future turns would likely be more than most tables might be comfortable with.