Interesting research, I thought someone here may find it interesting.

Has anyone seen a grid of knots while in communion with a plant teacher? That is what the early inflationary universe may have looked like...

Maybe, maybe knot.

Oh dear, Will I be able to control my knot jokes now?

I'm a frayed knot...

I don't think that would be the reason. You can knot 2D objects in 4D space.
Also the string theory computes with at least 10 dimensions.
But I think some limits that would need a 3D universe are more likely these instead of knots:
Cross-product that physics uses a lot (especially in electro-magnetic fields) can only work in 3D.
In 2D it can't happen and in 3D it would only work with 3 vectors instead of 2.
Orbits (solar systems and galaxies) couldn't form in 4D as in 4D there wouldn't be a single angular momentum to be conserved.
There could be much more, but this is all I could think of right now on top of my head.

The way I understand it they use two observations in their formulation:

(1) quarks are coupled by gluons. This system behaves like a string (which can be stretched and broken into more strings but never to a single quark/point).
(2) In the initial universe these strings expanded. They would have encountered resistance when expanding ONLY in 3D space (since they are only knotted in 3D. Therefore these strings blew up (expanded) the 3D space we live in know.

So you need both conditions (one from the particular particle fauna of our universe, and another from more general math that applies to all universes).

Your comment of 2D objects in 4D space does not invalidate their research, I think. Instead, it gives a recipe for a 4D universe and generalizes the theory: setup a universe with a bounded particle system that is 2D and cannot be broken into strings (e.g., a disk that upon expansion breaks up into other disks). When these fundamental particles systems expand during the big bang/inflation, they will tangle up and find resistance in 4D space since they are knotted there. The result will be that 4D space is expanded, and the resulting universe will be a 4D universe.

To test this we would need to setup a new universe having a fundamental 2D bounded particle system equivalent to our quark system. If the resulting post inflation universe is 4D, that would support this theoretical research. Sounds like an impossible experiment, but may be testable if you have a small solid state system/crystal that does contain 2D energy bound states and then allow it to expand with a lot of energy. You may get a very odd system with 4D internal properties.

Another way to think about this:

Imagine a 2D universe. There are other dimensions, but they are small and curled up. A being in this universe discovers quarks and their bound states wich behave like stings. This being puts a bunch of these strings in his lab and gives them a tremendous amount of energy. The strings will be mostly untangled as they mostly live in a 2D word, but some will be Tangled in the tiny 3D dimension. What will happen to these knots as the strings expand quickly? Would they expand the 3rd dimension creating a 3D "bulge" around the lab in this otherwise 2D universe?

Or more simple example:

Start with a flat table. Add grains of sand (point particles) and shake/stir to give energy: no new dimension shows up. Add cards (2D bound state) and jumble them up, they mostly remain flat on the table (again no new dimension shows up). Now add strings (1D bound state) and jumble them up. They knot up and grow in the 3rd dimension!

Now, based on the math that skr_nexus brought up, if one could give the cards enough energy without destroying them, they may knot up in the 4th dimension and expand it. That would be an interesting object...

Next time you shuffle a deck of cards, consider you may be tugging on the 4th dimension

3D knows showing up somewhere else: tangled magnetic spins of atoms that can live long enough to form a "skyrmion". They may be important to one day develop a practical nuclear fusion energy source.

Press release

Reference:

W. Lee, A.H. Gheorghe, K. Tiurev, T. Ollikainen, M. Möttönen, and D.S. Hall. Synthetic Electromagnetic Knot in a Three-Dimensional Skyrmion. Science Advances, 2018 DOI: 10.1126/sciadv.aao3820

The device they used to make it looks pretty much like a polywell, yet another interesting plasma fusion device based on a fusor device.