Alternating links and cobordism

A knot in mathematics is a closed loop in space - this is the same thing as a knot in a piece of string except we stipulate that the ends of the string should be joined together after tying the knot. This research programme will use 21st century mathematics to solve problems in knot theory that have defied solution for over a hundred years.Mathematical knot theory began in the 19th century as an attempt to compose a table of elements based on Lord Kelvin's theory of knotted vortices in the aether. The aether theory proved incorrect but started a rich mathematical study that now has many important applications. In particular DNA molecules exhibit knotting behaviour and the mathematical properties of the knots involved have important biological implications. In order to replicate, knotted DNA needs to become unknotted by a sequence of crossing changes . A crossing change or strand passage is when one strand of the knot is cut, and another strand passes through the cut which is then repaired.Edinburgh physicist P. G. Tait was the first to study these crossing changes, also in the 19th century. He defined a measure of complexity of a knot called the unknotting number, which counts how many crossing changes are needed to completely undo the knot. Computing these numbers is a notoriously difficult problem to this day; in fact there is no known algorithm for deciding if a knot has unknotting number equal to one.Knots are also used in giving mathematical descriptions of 3 and 4 dimensional spaces (or manifolds) such as the universe we live in. Another notoriously difficult and important problem is to decide if a knot is slice; thinking of time as the fourth dimension, a slice knot is a snapshot of a two-dimensional sphere in spacetime. (This problem is only around fifty years old.)In the last two decades of the 20th century new techniques pioneered by Donaldson, Witten and others transformed mathematicians' understanding of 4 dimensional geometry and topology. This new mathematical gauge theory was derived from the quantum field theories of theoretical physics.In the last ten years, a new version of mathematical gauge theory due to Ozsvath and Szabo has made major progress on problems in knot theory and 3-dimensional topology.Owens will combine gauge theory results of Donaldson with the new theory of Ozsvath and Szabo to attack the unknotting number and slice problems for a major class of knots known as alternating knots (these include well-known knots such as the granny knot and reef or square knot) This class of knots is known to be prevalent in knotted DNA. Part of the goal is to find a complete solution to the unknotting number one and slice recognition problems for these knots. Further crossing change information of interest to biologists will also be discovered as well as new insights into the mysterious mathematical nature of this very familiar class of knots.

The intended beneficiaries of this proposal include pure mathematicians working in topology and related areas, mathematical biologists studying DNA, students in mathematics, mathematics teachers, and the general public. The main impact of this research proposal will be knowledge in the form of scientific advance. This impact will be felt by academic mathematicians and mathematical biologists within five years. Educational and outreach impact is also expected within five years. There is a possibility of indirect benefit to biomedical research over a longer period. The primary academic beneficiaries will be mathematicians working in the field of topology in the UK, and overseas (including the USA and various EU institutions). The research will provide new techniques and information to mathematical biologists studying topology of knotted DNA molecules, and the action of specific proteins which change this topology. In the longer term this may conceivably lead to benefits for medical research. Another benefit of this project will be training of researchers. In each year of the project the PI will engage a summer research student to help with the project. They will assist the PI with various computer programming components and will gain valuable research experience and training. This project will have impact in education and outreach through the development of computer animations illustrating slice disks in the four-dimensional ball. They will be of interest to experts but will also provide a valuable resource for maths teachers, students and members of the public to get insight into topology in four dimensions.