Knot Math: Unraveling a Complex Theory
Don’t worry if you aren’t a fan of calculus or if algebra isn’t your thing, this article is about knot math. Knot theory is a popular subfield of mathematics with new and exciting applications. Here, we will briefly describe the application of knot theory to building a quantum computer and to the study of DNA, but first we need a bit of background information.
What is a knot? Take a piece of string, make a knot in it, and glue the ends together. While this may not seem like such a big deal, to mathematicians this last step is absolutely crucial. An important characteristic of a knot is that it must be closed. The simplest of all knots is called the “unknot,” which is a circle. Another way to form knots is to take several pieces of string and braid them – as if you are braiding a young girl’s hair – then connect the tops of the strings to their respective bottoms. This will either give you a knot or a link, which is a collection of knots that do not intersect, but may be linked together. For example, if you have two straight strings without any crossings and you connect the tops to the bottoms, you will construct two unlinked unknots; however, if you cross the right string over the left string three times and then connect the tops of the strings to the bottom of the strings, you will have formed a knot called the “trefoil.” Try it. This is the simplest nontrivial knot. An important mathematical theorem states that any knot can be formed by connecting the ends of a braid. Thus, braids play a key role in knot theory, which brings us to our first application.
As computers continue to get smaller and faster, they are approaching physical limits, propelling the world toward quantum physics. In the 1990s, Peter Shor, Ph.D., currently a professor of applied mathematics at MIT, showed that quantum computers would have the power to factor large numbers quickly and in doing so would break cryptographic codes, making commonly used electronic communication not secure. Secure electronic communication is frequently taken for granted when purchasing items with credit cards or conducting online banking. The biggest problem with a quantum computer is that no one knows how exactly to build or even maintain one. There are several approaches that are currently being attempted, but each has its own unique limitations. One method to construct a quantum computer is based on the idea that the quantum bits can be encoded into small particles called “anyons.” Braids can model the trajectory of these particles over time. This is the basis of what has been called a topological quantum computer since knot theory is a part of mathematics called “topology.”
Another application of knot theory is utilized in understanding the complex and dynamic structure of DNA. Strands of DNA are often bunched tightly together and can become entangled. Knot theory can be used to study the biological process of DNA recombination where a crossing of two strands may switch or break apart and recombine at different ends. Telling two knots apart is extremely difficult. As a result, mathematicians use invariants to help. A knot invariant is something that is assigned to each knot and remains unchanged even if the knot looks different. For example, some of the most well-known knot invariants are polynomials. Each knot is assigned a specific polynomial. If the polynomials are different, the knots are different, but if the polynomials are the same, they may or may not be the same knot.
We have described only two of the numerous applications of knot theory to physics and biology. There are many more deep connections to complex areas such as protein folding, synthetic chemistry and statistical mechanics. The interested reader is encouraged to explore our references below. One of the benefits of studying knots is that all you need is a few pieces of string to get started!
• Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society.
• Collins, Graham (April 2006), “Computing with Quantum Knots,” Scientific American.ere