The Knotted Molecule

Enormously long chainlike molecules (long in relation to their breadth) have been discovered in living organisms. The question has arisen: Can such molecules have knotted forms? Max DeLbriick of the California Institute of Technology, who received a Nobel prize in 1960, proposed the following idealized problem:

Assume that a chain of atoms, its ends joined to form a closed space curve, consists of rigid, straight-line segments each one unit long. At every node where two such "links" meet, a 90-degree angle is formed. At each end of each link, there-Fore, the next link may have one of four different orientations. The entire closed chain could be traced along the edges of a cubical lattice, with the proviso that at each node the joined links form a right angle [see Figure 10]. At no point is the chain allowed to touch or intersect itself; that is, two and only two links meet at every node.

Figure 10

Figure 10

Example of a 13-link chain

Example of a 13-link chain

What is the shortest chain of this type that is tied in a single overhand (trefoil) knot? In the answer I shall reproduce the shortest chain Delbrück has found. It has not been proved minimal; perhaps a reader will discover a shorter one. (I wish to thank John McKay for calling this problem to my attention.)

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