Knots, trivial and otherwise

Crocheted theta curve sea creature by Jake Wildstrom

I think about knots a lot these days, and I think about how complicated those knots are. My work involves knots, and knots within knots. The fanciful sea creatures I’m crocheting are a variant of knots called theta curves.

Crocheted theta curve sea creature by Jake Wildstrom
Early mock-ups of the crocheted theta-curve sea creatures. This
particular creature is known as a theta cinquefoil.

Knots with a mathematical twist

While knots, in the mathematical sense, are tangled loops, theta curves are tanglings of a shape like the Greek letter θ. These sea creatures, however, are not only knotted in their overt shape, but also in their construction, for crochet fabric is built up by using a hook to cross the yarn over itself, building tangles whose complexity increases with each stitch made.

Complex or knot

This complexity of crocheted work, and likewise of knitted work, is only truly realized in a final, extraordinary step. While the conventional notion of a knot is of something done to a strand, mathematical knots are closed loops. The artist, during the work, can become part of that loop: taking the ball of thread in one hand, and the tail of the work in the other, they become a closed loop which contains their work. How complicated, we might ask, is this loop?

There are several ways of quantifying how simple or complicated a knot is, but most of these metrics believe that a knot is identical, both in identity and in complexity, to any knot you can transform it into by performing the Reidemeister move. Every knit and crochet stitch is actually a Reidemeister move (mostly the Type II moves), so when a work is in progress, everything being done, from a mathematical perspective, is adding no complexity at all, and the work is, mid-construction, still equivalent to a simple closed loop with no knots in it at all, which is mathematically known as the unknot.

The three Reidemeister moves by Jake Wildstrom
The three Reidemeister moves, identified both by the numbering originally given, and their common names.

From trivial to extraordinary

Knitters and crocheters are all-too-well aware of their work’s underlying triviality. A sharp tug on a crochet work, or a dropped stitch in knitting, will impel the work to unravel itself. It’s only in the last step that these yarncrafts suddenly jump, in a mathematical sense, from the trivial to the extraordinary. When the yarn is cut, and the loose end drawn through the last stitch, then the knot formed by the work and the crafter’s body is no longer trivial, but one whose many, many crossings cannot be undone with Reidemeister moves. By one standard of measurement, the work is still very simple—the unknotting number is the number of times the yarn would need to be crossed over itself to become trivial once more, and that number is only 1—but by almost any other standard, even a simple crocheted or knitted work is a knot of tremendous complexity.

The tremendous complexity of the ostensibly trivial is also familiar to yarncrafters in a different phenomenon, the so-called “yarn barf” frustration when a center-pull ball abruptly releases a bolus of tangled yarn onto the working end of the project. Because of the way center-pull balls are wound, this tangle is always trivial in the mathematical sense, in that it could be completely straightened without having to pull either the ball or the work through the tangle. A theoretical triviality is no help to the crafter who wants to know how to get the yarn running smoothly again, though!

Algorithms with knots

How to disentangle entangled yarn, or even how to recognize, without outside context (as we do have in the case of center-pull balls) that a strand can be disentangled, is a question in the field of algorithms. This particular algorithmic problem is known as the unknotting problem, and not much is known about it. There’s a scaling level of difficulty for algorithmic problems, but unknotting is only very vaguely placed within this scale: there’s a family of algorithmically easy problems called “P”, a family of tractable but seemingly difficult problems known as “NP-complete”, and a larger genus containing both P and NP-complete, known as “NP”. The unknotting problem is only known to belong to NP, which doesn’t tell us much.

As stitches build up on a work, these ideas of complexity and triviality are always present. The work, even as it gains in apparent complexity is a triviality constantly threatening to materialize, and the yarn barfs’ apparent complexity stubbornly resists being exposed as trivial. Even the question of what is trivial and how we can show it proves, in the end, to be itself, on a different plane, a very complicated question.

For a deeper look into Knot Theory

Published by Jake Wildstrom

Jake Wildstrom is an associate professor in the mathematics department of the University of Louisville, in Kentucky. He received his Ph.D. from the University of California at San Diego in 2007. His academic specialty is combinatorial optimization, specifically with regard to communication and transportation networks. Jake has participated in several Bridges Math & Art conferences, and his artistic work is primarily in the field of crochet. His mathematical explorations into crochet have included theoretical designs and practical implementations of intermeshed crochet grids, self-similar and fractal designs, center-worked irregular polygons, braids and knots, and the controlled use of randomness as a design element.

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