Hamiltonian knot projections and lengths of thick knots

Abstract

For a knot or link K, L( K) denotes the rope length of K and Cr( K) denotes the crossing number of K. An important problem in geometric knot theory concerns the bound on L( K) in terms of Cr( K). It is well known that there exist positive constants c 1, c 2 such that for any knot or link K, c 1·( Cr( K)) 3/4⩽ L( K)⩽ c 2·( Cr( K)) 2. In this paper, we prove that there exists a constant c>0 such that for any knot or link K, L( K)⩽ c·( Cr( K)) 3/2. This is done through the study of regular projections of knots and links as 4-regular plane graphs. We show that for any knot or link K there exists a knot or link K′ and a regular projection G of K′ such that K′ is of the same knot type as that of K, G has at most 4· Cr( K) crossings, and G is a Hamiltonian graph. We then use this result to develop an embedding algorithm. Using this algorithm, we are able to embed any knot or link K into the simple cubic lattice such that the length of the embedded knot is of order at most O(( Cr( K)) 3/2). This result in turn establishes the above mentioned upper bound on L( K) for smooth knots and links. Moreover, for many knots and links with special Hamiltonian projections, our embedding algorithm ensures that the bound on L( K) can be of order O( Cr( K)). The study of Hamilton cycles in a regular knot projection plays a very important role and many questions can be raised in this direction.