The PIs propose to investigate the space of very large knots (with thousands of crossings) through a symbiosis of theoretical research and computation. The problems raised here are motivated by the applications of knot theory in chemistry, physics, and biophysics. The main goal is to develop and implement new algorithms that sample the space of large knots and that are capable of embedding large knots tightly or semi-tightly in the simple cubic lattice. In this research, theoretical work will enhance the algorithms to be developed and the empirical results will support the theoretical approaches. The proposed research project consists of several inter-related objectives: developing fast algorithms capable of generating representative samples of large knot diagrams, developing better algorithms capable of embedding large knots tightly in the simple cubic lattice, and developing new theoretical approaches to improve the upper and lower bounds of the ropelength in general or for special knot classes such as alternating knots. These objectives are difficult and challenging. For example, the distribution of large knots is unknown, determining the crossing number of large non-alternating knots is known to be NP-hard, and theoretical results concerning large knots are scarce in general. Concretely the PIs expect to sample the space of large knots using and comparing three approaches: first applying uniform prefix vectors to generate large Hamiltonian prime knot diagrams; second, adapting a method based on uniform random polygons to sample large prime knot diagrams; finally, using graph tensor products to construct large non-alternating knots whose crossing number can be approximated by computing the breadth of the Jones polynomial (via the Tutte polynomial); sampling such non-alternating knots allows a comparison with alternating knots of similar size. Furthermore, the PIs will work on developing a more efficient embedding algorithm by extending the constructive proof of two of the PIs regarding the embedding length of closed braids to general knots. This approach also aims at improving the general upper bound on the ropelength of knots from the current bound of crossing number to the power of 1.5.
The main subjects to be studied in this proposal are large physical knots, i.e. large knots that can actually occur in the real world. Examples of such occurrences are long, knotted polymer chains or circular DNA. The problems raised in this proposal are motivated by the applications of knot theory in chemistry, physics and biophysics. For example, DNA knots formed under extreme conditions of condensation, such as those found in bacteriophage P4, can be quite large. Such large and tightly packed circular DNAs are difficult to analyze experimentally. Theoretical results or computational simulations on such systems would be of great help. Yet theoretical studies on large knots are scarce. The proposed project aims at gaining more knowledge about large knots: How much space is needed in order to pack certain large knots? How efficiently can knots be packed tightly? Is there a difference between packing a complicated knot in comparison to packing a simple knot? What role does the topology (shape) of the knot play? The PIs intend to develop computer programs that can generate large knots and pack them tightly, based on theoretical results and algorithms they have developed in the past. Empirical data can then be gathered through the repeated applications of these programs. The proposed activities may have significant implications in DNA research, polymer science, and other sciences. The activities will result in tools for researchers to compute various geometric and topological characteristics of the large knots they encounter in their field and thus help them to better understand biological and physical systems where large knotted molecules occur.
