Telekinesis can be Twinned, part 1
Contrary to the other answers so far posted, Telekinesis can be Twinned. Bigby's Hand and Eyebite cannot for the reasons the other answers have stated.
But before I go into why, let me point at examples of spells that can be twinned. This will make sense in a minute.
Hex can be Twinned
The Warlock spell Hex fulfills all the requirements of Twin Spell.
- It targets one creature
- It is incapable of targeting more than one creature at any level of the spell
- It does not have a range of Self
There is a caveat, though: on subsequent rounds, after the target's death, you can use a bonus action to move the hex from the original target to a new one.
Point 2 may be where the disagreement comes in, but I would say most people will agree that Hex can be Twinned. The reason you can Twin it is, at the time of casting this spell, you cannot target more than one creature. You cannot do so no matter the level of the spell slot used. And you cannot move the spell until the target dies, anyway.
Hunter's Mark can be Twinned
This spell is similar to the previous one, but has an important distinction: it does not need the target to die before you can transfer the spell onto another creature. But, Hunter's Mark fulfills all of the same checkboxes as Hex:
- It targets one creature
- It is incapable of targeting more than one creature at any level of the spell
- It does not have a range of Self
And so, if you accept Hex can be Twinned (which I believe most people would), then you must accept that Hunter's Mark can be Twinned as well.
Ice Knife can be Twinned
Now we are getting into weird territory. What I have shown is you can twin a spell as long as it cannot target more than one creature at the time of casting. Ice Knife has a range of 60ft and has the following effect:
You create a shard of ice and fling it at one creature within range.
Make a ranged spell attack against the target. On a hit, the target
takes 1d10 piercing damage. Hit or miss, the shard then explodes. The
target and each creature within 5 feet of the point where the ice
exploded must succeed on a Dexterity saving throw or take 2d6 cold
damage.
You first target one creature, which you make an attack roll against. This target then takes 1d10 piercing damage. If the spell ended there, then we can unambiguously say that this spell is eligible for Twinned Spell.
However, it has an extra effect: the shard you threw will explode and damage enemies in an area around your initial target. So, where does this leave the spell with regards to Twinned Spell?
Let us fall back to Hex and Hunter's Mark: both spells are eligible for Twinned Spell, but both can affect more than one creature over their durations -- but they can only ever affect one creature at a time. More importantly, they can only affect one creature at the time of casting.
Now, Ice Knife has a duration of Instantaneous. That means, as soon as you conjure and throw the ice knife, that is the effect of the spell that directly results from your casting. And at the time of casting of Ice Knife, you choose one target, and the Instantaneous nature of the spell means your choice was only made so far as to the target of the knife you threw. The ice simply explodes as a further effect of the spell -- but by the time the knife explodes, your casting of the spell is over.
This is essentially the reverse logic of why Bigby's Hand or Maximillian's Earthen Grasp cannot be Twinned -- while they can only target one creature, they first create an effect and then target one creature. Ice Knife provides the reverse -- it targets one creature and then creates an effect.
Now, what of PHB pg 205. that reads:
Many spells specify that a target can make a saving throw to avoid some or all of a spell's effects
Well, this is true for Disintegrate, Fireball, etc. But it doesn't explicitly say "all targets that make a saving throw due to a spell's effects are to be considered as targets of the spell". This means, the creatures making saving throws do not necessarily have to be treated as targets of the spell, if this is the only line we are using as basis.
So, if you believe that Bigby's Hand or Maximilian's Earthen Grasp cannot be Twinned, then you must believe the reverse logic, and Ice Knife can be Twinned.
Telekinesis can be Twinned, part 2
Now we come back to Telekinesis. We so far know that, as long as the spell only targets one creature upon casting, it can be Twinned (see: Hex). This is true even if you can transfer the effect via a bonus action (see: Hunter's Mark).
Telekinesis, now, explicitly allows you to choose your target. When you target a creature, this is the pertinent clause:
Creature. You can try to move a Huge or smaller creature. Make an ability check with your spellcasting ability contested by the
creature’s Strength check. If you win the contest, you move the
creature up to 30 feet in any direction, including upward but not
beyond the range of this spell. Until the end of your next turn, the
creature is restrained in your telekinetic grip. A creature lifted
upward is suspended in mid-air.
This is not negated by the fact you can use your action to move the spell to a different creature, or even to affect a different object.
But Wait! Telekinesis Creates An Effect First
The same reasons why Bigby's Hand and Maximilian's Earthen Grasp may be used to say Telekinesis is ineligible -- they both create an effect first which then targets a creature. Strictly speaking, this disqualifies them from Twinned Spell.
However, I contend that Telekinesis does not share the same limitations.
Telekinesis reads, at the first sentence:
You gain the ability to move or manipulate creatures or objects by thought.
This sentence does not say anything about the target of the spell. While it may seem to be the effect the spell creates, that does not mean you have determined the target of this spell already.
Yes, this is an effect of the spell, but it is also not the only effect. This is an effect. One of many, and this particular effect is silent on who the target is.
Note this sentence is not synonymous with:
You are the target of this spell, and therefore gain the ability to move or manipulate creatures or objects by thought.
In fact, it is just as valid to interpret that sentence as:
While you gain the ability to move or manipulate creatures or objects by thought, you are not the target of the spell.
For reference, see Warding Bond. Warding Bond is a single-target spell which affects the caster, but targets a different creature. This allows the same caster to cast Warding Bond multiple times on different creatures, without breaking the clause of that spell that you cannot target a creature already affected by Warding Bond.
That is to say, there is another spell that affects the caster, but does not target the caster. This precendent lends credence to the fact that a spell that affects the caster does not necessarily target them.
Contrast this to Bigby's Hand:
You create a Large hand of shimmering, translucent force in an unoccupied space that you can see within range.
As well as Earthen Grasp:
You choose a 5-foot-square unoccupied space on the ground that you can see within range.
It is clear in both spells that they first create effects, which, as a consequence, allow you to target other creatures via that primary effect. The same cannot be said for Telekinesis in a clear-cut way.
Now, the 2nd and 3rd sentences for Telekinesis reads:
When you cast the spell, and as your action each round for the duration, you can exert your will on one creature or object that you can see within range, causing the appropriate effect below. You can affect the same target round after round, or choose a new one at any time.
This is a bit more explicit. We said that for Hex, Hunter's Mark, and Ice Knife, you can Twin all those spells because, at the time of casting, you can only target one creature. Now, let us postulate that the phrase "at the time of casting" implies the same thing as "when you cast the spell" -- that is, both phrases, in plain English, refer to the instant the spell is cast.
Now, Telekinesis says that the instant the spell is cast, you may target one creature. It is quite explicit: at the moment of casting, you can choose to target one creature. If this is what you do, then it is eligible for Twinned Spell. If you target an object, then it is not eligible. And remember, the fact you can target either a creature OR an object does not disqualify it -- what matters is, at the time of casting, it cannot target more than one creature.
Stacking Telekinesis On One Creature
But wait! Does this mean you can initially cast Telekinesis on two creatures, and then move the two effects to one creature?
Yes! Yes, you can. But they are affected by only one spell effect. This is due to PHB pg 205, which reads:
Combining Magical Effects
The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect—such as the highest bonus—from those castings applies while their durations overlap.
For example, if two clerics cast bless on the same target, that
character gains the spell’s benefit only once; he or she doesn’t get
to roll two bonus dice.
So one creature affected by two Hexes, or two Hunters' Marks, or two Telekinesis, will only be affected by the spell once. This makes sense if you imagine two opposing Wizards casting Telekinesis on the same target in a tug-of-war: only one Wizard will succeed in controlling the target, while the effect of the less potent spell is temporarily suppressed.
Houseruling And Does It Break The Game?
Now, should you disallow your Sorcerer from Twinning these spells? Well, that depends on the balance you want to strike in your game. If you see that it is going to break many of your encounters, then rule they cannot be twinned. If, on the other hand, it doesn't really change much and you can design with this in mind, then allow it by all means. After all, getting Twinned Spell is an investment from the player, and being able to hold two Fire Giants with your arcane prowess is nothing short of cool -- and we value Rule of Cool around these parts. :)