This is a CARGO incubation award made under solicitation www.nsf.gov/pubs/2002/nsf02155/nsf02155.htm. New applications of knot theory in biophysics and biochemistry have focused interest on knotted molecules, that is, on physical knots of volume-occupying nature and particular geometric shapes. A physical knot is modeled as a smooth closed curve with certain thickness. The thickness of a smooth knot can be thought of, intuitively, as the radius of the rope with which the knot is tied as tight as possible. In mathematical terms it is the largest embedded normal tube around a smooth knot that does not intersect itself. The ropelength of a (thick) knot is the quotient of its arc length over its thickness. The focus of this proposed work is the challenging problem of finding good estimates of the minimum ropelengths of various knots. For knots with small crossing numbers, computational methods exist which estimate their ropelengths. The techniques used to obtain these estimates cannot be extended to very large knots. Thus very few computational results are available for knots with large crossing numbers. It is known that for any nontrivial knot, its minimum ropelength is bounded above by a constant times its crossing number squared. The PIs have recently improved this bound to a constant times the crossing number (of the knot) raised to the three half power. However, the PIs suspect that for most knots, their ropelengths are bounded by their crossing numbers (times a constant) or less. So there seems to be a gap between the proven bound and the actual minimal ropelengths of various knots. In this work, the PIs will develop a computer program that is capable of computing a close estimate of the minimum ropelengths of knots with large crossing numbers. This computer program will make use of the algorithm used in establishing the upper bound of the three half power mentioned above.
Some research in biophysics, chemistry and physics deals with long strings of molecules that are knotted, for example, circular DNA or polymer chains. The type of knotting often influences the properties and behavior of the molecules. To better understand the properties of such molecules, this project proposes to examine the relationship between the complexity of a knot and its physical length. This relationship can be formulated into the following question: If one is given a rope of certain radius and wants to tie a certain knot, how long does the rope have to be? For small knots, one can get a rough idea by simply tying the knot and measuring how much rope it took. However for very large knots that is not practical since it is not clear how to tie a knot with rope in an optimal way. The proposed project involves the creation of a computer program that - for many knots -checks lots of different ways how the same knot can be tied, in order to find a close estimate to the minimum length of rope. The results of the computer program will be used to hypothesize a general relationship between the complexities and the lengths of knots.
