When I hear the words geometry and knot theory together, my first inclination is to think about the canonical geometries (usually hyperbolic) on knot complements. However, one can also consider how the knot sits with respect to the Euclidean geometry in **R**^3: writhe, energy, etc. Rope length falls into this second category. Think of our normally width-less knots as if they were in fact made of reasonably thick rope. In other words, we want to embed it in space so that if we choose any two points and take perpendicular disks of radius one at these points, the interiors of the disks will be disjoint. The *ropelength* of a knot (or rather an isotopy class of knots) is the minimum length over all representatives with this property.

It’s not too hard to find a length 2pi embedding of the unknot with this property and takes only a little more work to show that 2pi is in fact the ropelength of the unknot. That’s the point where things cease to be easy. Computer modeling has produced an embedding of the trefoil with length around 16.3 and this seems to be the best one should expect. This gives us an upper bound on the rope length of the trefoil. A good lower bound on the other hand (i.e. a lower bound anywhere near the upper bound) has proved extremely difficult to find.

Cantarella, Kusner and Sullivan [1] managed a bound of around 10.7 in 2001. (They also proved the existence of minimal length embeddings.) Yuanan Diao [2] bumped the bound up to 12 (i.e. a foot) in 2003. That put the lower bound at under three quarters of the expected value. The following year Denne, Diao and Sullivan [3] got the lower bound up over 15.6, about 96% of the expected value.

Based on an ArXiv search and a Mathscinet search, there don’t seem to be any results since then that improve the bound for the trefoil or give a bound for other knots. (Though it is possible I’m not searching for the right keywords.) The bounds above seem to be for any non-trivial knot. That is, they don’t seem to use anything about the topology of the trefoil other than that it’s not the unknot. It would be interesting to see if there are any techniques that use the specific topology of a given knot. Then one might be able to get a lower bound for, say, the figure eight that’s higher than that of the trefoil. I’m not saying that this would be an easy thing to do (in fact it’s probably very very difficult), but it would be pretty interesting.

[2] Yuanan Diao, The lower bounds of the lengths of thick knots. J. Knot Theory Ramif., 12:1,

2003, pp 1–16.

So what happens if you consider rope length (as a function of the rope’s thickness, as the thickness gets large) in 3d hyperbolic geometry?

Comment by D. Eppstein — June 5, 2008 @ 8:42 pm |

I guess there are two ways to interpret that. The directly analogous problem would be to think of the knot as sitting in hyperbolic 3-space, in which case I don’t really know what happens… Would there still be a linear relationship between length and thickness?

A less direct analogy would be to consider the complete hyperbolic structure on the knot complement, in which case the knot sits at the end of a cusp. You can then extend a neighborhood of the cusp (the image of a horoball) into the interior until it becomes tangent to itself. You can’t measure the diameter of this neighborhood, but you can measure it’s volume. As I understand it, many of the lower bounds on volume for cusped hyperbolic 3-manifolds start with a construction along those lines (though some readers of this blog know more about this than I do). I wonder if there’s any relationship between this volume and the rope length. (Seems unlikely…)

Comment by Jesse Johnson — June 6, 2008 @ 6:32 pm |