New homotopy-type invariants of knots.

To a mathematician knots are essentially the same as they are to anyone - tangled up pieces of string. Studying when two knots are different turns out to have deep implications for a wide range of mathematical areas. In this project we are developing a new way to distinguish between two knots and we hope that our research will lead not just to a better understanding of knots, but also to a better understanding of weird 4-dimensional spaces.

This hope comes from the fact that one can consider knots sitting in 3-dimensional space as living on the boundary of 4-dimensional space. When one pulls the knot through the 4-dimensional space, analyzing what sort of surfaces you can produce tells you about that 4-dimensional space. Our way of distinguishing between knots should have something to say also about these surfaces, and hence about 4-dimensional spaces.

Knot theory and, more generally, low-dimensional topology has found applications into studying the tertiary structure of DNA. The tertiary structure refers to the large-scale behaviour of DNA, in which the double-helical structure is not directly seen, but rather the strand of DNA is considered as a filament that might wrap around itself and get knotted up.

Results from low-dimensional topology and knot theory have often found direct relevance to this area of biology, especially in evaluating which topological knot and link types could arise within the cell. The knots for which computation of our invariants may be tractable are exactly those knots which admit a "supercoiled" structure. This structure is common in knotted DNA and is caused by the overtwisting of the strands of the double helix around each other. Hence there is a hope that our invariants (and the knot homologies whose structure they illuminate) might have an impact in the biological sciences.

Another reason for this hope is that our strengthening of the slice genus bounds mentioned in the proposal should give new information on the number of crossing changes needed to get from one knot to another. Topoisomerases act exactly by changing crossings between two strands of DNA - they tend to act to unknot knotted DNA in order that it can replicate. Hence this "crossing-change distance" is a very important metric for topological biologists and new information about it should be welcome.

Of course, such impacts, while not certain, have the potential to be large. A better understanding of the tertiary structure of DNA may help analysis and treatment of genetic disorders or have applications to the fight against cancer.