‘And now we work’, said my supervisor, the week after I got accepted for SRS. This suited me fine, it’s sad not doing maths. Also, I study part-time, because disabilities. Also, I was already teaching myself Reidemeister moves with a ribbon and photos of knot transformation stages. Hint: it’s easier to compute knots by thinking and drawing alone. Ribbons aren’t infinitely elastic when you most need them to be and suddenly fall off the table. But at first, drawing knots is difficult. Without the ipad, my cats would be big balls of eraser dirt now.
I am interested in topology, and curious how it applies to biological structures, i.e., how mathematical knot theory models DNA processes. Natural shapes fascinate me. Learning about knot theory began with studying parts of Kauffmann’s Knots and Physics (2004) to acquire some knot theory background. As DNA applications use rational tangles, I undertook a literature review about (i) the mathematical structure of rational tangles, and (ii) their current applications in biology. According to Conway’s Theorem, two rational tangles are isotopic if and only if they have the same fraction invariant p/q, implying a bijection between rational tangles and rational numbers. How is this possible, when the arithmetic of rational tangles is not fully commutative, and simple additions can lead to a non-rational tangle? Hence I focused on the algebraic structure of rational tangles, working on problems, examples, counterexamples, and sometimes, physical models.
My main focus was Kauffmann and Lambropoulou’s (2004) proof of Conway’s Theorem, a labyrinth of re-jigged simple puzzle pieces. Working through details, I learned how strategically and rigorously bringing the notation of different fields ever closer together creates new connections. The shared underlying structure turned out to be the modular group, which ultimately explains the curious tangle arithmetic as matrix multiplication.
The DNA application already relies on topological Dehn surgery, which in turn creates a (p,q) lens space. Rational tangles also relate to Pauli matrices, elliptic functions, and planar graphs. I hope to do further research into some of the deeper structures involved here. Rational tangles are a tool in different areas, which might benefit from stronger connections.
Come presentation time, my dream to show how half of maths is entangled with my topic met reality. Ouch! Only so much room on a presentation slide, and not all my speculations make sense. Drawing nice knots is like calligraphy and needs practice. It was worth it. Also, I presented without imploding from nerves.
I enjoyed the conference immensely. I can’t wait to entangle the rest of mathematics with my ideas on … we shall see.
References
Kauffman, LH 2001, Knots and Physics, 3rd edn, University of Illinois, Chicago.
Uli Krahn
The University of New England
