Narrow Results

The first author defined a well-order in the set of links by embedding it into a canonical well-ordered set of (integral) lattice points. He also gave elementary transformations among lattice points to enumerate the prime links in terms of lattice points under this order. In this paper, we add some new elementary transformations and explain how to enumerate the prime links. We show a table of the first 443 prime links arising from the lattice points of lengths up to 10 under this order. Our argument enables us to find 7 omissions and 1 overlap in Conway's table of prime links of 10 crossings.

The goal of this paper is to tabulate all genus one prime virtual knots having diagrams with ≤ 5 classical crossings. First, we construct all nonlocal prime knots in the thickened torus T × I which have diagrams with ≤ 5 crossings and admit no destabilizations. Then we use a generalized version of the Kauffman polynomial to prove that all those knots are different. Finally, we convert the knot diagrams in T thus obtained into virtual knot diagrams in the plane.

We tabulate all prime alternating knots through fourteen crossings. This provides the first list of 14-crossing prime alternating knots as well as the first confirmation of Thistlethwaite's tabulation of 12- and 13-crossing alternating knots.