Color-changing fibers help unravel a knotty problem

If you’ve ever tied your shoe in a hurry, you know that not all knots are equal. Some knots are stronger than others. And scientists have struggled to explain why. Now that’s changing, thanks to some color-changing fibers and math. A research team developed a few math-based rules that can describe knots’ relative strength based just on their topology. That refers to the geometry of how the knot is tied.

Vishal Patil is an applied mathematician at the Massachusetts Institute of Technology in Cambridge. He was part of a team that tackled the knotty problem. “Despite the fact that [knots] have been around for thousands of years, not much is known about why they work the way they do,” he says.

Patil and his colleagues started with very simple knots. Each was tied with a single fiber. And each was made with special fibers — ones that change color when they are stressed. The fibers’ different hues revealed areas of greater and lesser strain within a knot. The team also created computer models to simulate the stress those fibers had encountered. Patterns of strain in these knotted fibers matched well with what the computer had predicted, the researchers found.

What’s more, those strain calculations let the researchers estimate the relative strength of different knots. Patil’s group shared its new findings January 3 in Science.

Next, the team used what the computer had predicted to calculate the relative strength of more complex knots. For that, they used knots known as bends. These connect two separate pieces of rope.

Patil’s group now reports that just three features could explain a knot’s strength. First, the more times the strands cross, the stronger the knot. Any twisting of strands as they cross one another also plays a role. If the strands twist in opposite directions, the twist balances out — and that locks the knot into place. Finally, if neighboring strands slide in opposing directions as a knot is tightened, that also strengthens the knot.

The rules predict only the relative strength of each knot — that is, whether one knot is stronger than another. A knot’s overall strength would depend on other factors. These might include the type of rope or fiber used to tie the knot.

Still, the results help explain why some knots stay tied better than others. Consider the granny knot. It is notorious for causing loose shoelaces. The square knot looks similar but has a balanced twist. And that makes it stronger. In contrast, the granny’s twist is unbalanced — and that could really trip you up.