Why Knots?

by Greg Buck

Knots are fun. People smile when they talk about knots. Children seek them out on their own --- consider friendship bracelets, hair braids and scouting merit badges. Knots are ubiquitous. Every child or adult has some practical experience with knots. Knots are practical. This is, after all, why knots are so well known in the first place. Let knotting include braiding, weaving, and tangling, and see that we tie our shoes, braid our hair, lash trunk lids to bumpers and disentangle extension cords. We weave rugs, knit sweaters and tie on fishhooks. Knots are also cultural and artistic expressions. Every culture on the planet (and likely any other planet in this universe) know knots, and in every culture they have taken on a cultural role inspired by their obvious practical significance. There is the true lovers knot, the gordian knot, the Celtic art knots, Norse mystical knots, the braiding of hair, the marlinespike of the sailor.

Knots are also of fundamental mathematical and scientific importance. In mathematics, the understanding of knots is essential to the understanding of the nature, the topology and geometry, of three dimensional space. These areas of study have been some of the most fruitful of modern mathematics. Vaughn Jones won the Fields Medal (the closest mathematical equivalent to the Nobel Prize) for his work in knot theory, the work of several other recent winners includes or is related to knots --- we mention Bill Thurston and Ed Witten. In general, knots are to mathematics as white rats are to biology: of intrinsic interest and also a magnificent laboratory for making discoveries and developing tools that extend to other areas. For example, knotting and tangling are essential phenomena in the process of DNA replication, and the analysis of this phenomena has been the subject of a dynamic and productive collaboration of mathematicians and microbiologists. Other applications include electromagnetism, fluid mechanics, physiology, cosmology, and celestial mechanics.

The bridge between these two views of knots is elementary knot theory. And what is truly remarkable is just how accessible elementary knot theory is. Many children learn basic patterns of knotting and braiding and weaving before they learn arithmetic. Most children learn to tie their shoes before they learn to read. Our direct experience is that preschoolers can mimic braiding and weaving patterns and count crossings in planar diagrams --- the experience with braid patterns is echoed by recreation leaders and camp counselors. The elementary study of knotting and tangling utilizes several of the most important mathematical conceptual tools. We mention three. The first is pattern recognition and classification of forms. In fact the classification of knot types is in some ways the chief aim of mathematical knot theory. The second is algorithmic thinking. This is particularly apparent in the mathematics of braids and weaving. (Many consider the Jacquard looms to be the first computers). The third is spatial understanding. Real knots are inherently three dimensional objects whose properties can be discerned by considering their three dimensional abstract forms.

There are several other qualities of knots that make them an attractive medium for mathematics education. Knotting and tangling are tactile and visual phenomena. One can easily imagine students who would be fascinated by knotting who would not be otherwise interested in mathematics. This literal ``hands on'' quality is rare in mathematics.

Knots are readily available and easy to distribute. The materials required are nothing more than everyday clothesline, or other rope, and pencil and paper. Therefore exercises and activities can easily be distributed by web page or print media, and these can easily be implemented in the classroom, home, or other venue of study. Moreover, with these simple materials, students of any age or ability can conduct experiments which are not simplistic or artificial models, but rather the real thing. Mathematics has been called the science of patterns, and knots are the most robust of patterns --- requiring only a length of string or rope for their creation and sustenance.

Knots are multigenerational, another rarity in the mathematical world. Basic knotting and braiding is exactly the sort of knowledge which is often passed between generations in the home, and this shared interest is thought to be an important aspect in science education.

Finally, the advent of computer technology has meant a great deal to the study of knots. In fact knots are a wonderful link between the cultural and scientific ways of knowing. With new scientific approaches to knots, especially those utilizing the computational and visualization strengths of new computers, science is now tackling facets of knotting and tangling that have been understood (or at least approached) culturally for decades, centuries, or millennia. The intersection has provided a dynamic boost to both approaches.