Exhaustive Generation of Genus-One Knot and Link Diagrams via Maps on the Torus

Abstract

We present an algorithmic framework for the exhaustive generation and tabulation of knot and link diagrams on the thickened torus T^2 x I, based on the theory of maps on surfaces. Cellular 4-regular torus projections are encoded by permutation pairs (alpha, sigma), and unsensed equivalence classes are enumerated completely and without duplication via canonical representatives. Crossing assignments, local diagram-level reductions, and the generalized Kauffman-type bracket are formulated entirely within the same permutation model. The pipeline is validated against published genus-one classifications for crossing numbers N <= 5 and then extended to N = 6, 7, 8, producing, to our knowledge, the first complete genus-one tabulation at these crossing numbers under the stated comparison conventions. The resulting dataset contains more than 33,000 knot and link types. Besides the tables, the computation yields proved structural facts, including a parity statement for the a-span of the bracket and a sharp upper bound N-1 for the number of bigon faces in a 4-regular torus map. It also suggests several conjectures, among them a formula for the maximum number of straight-ahead components, the absence of equi-quadrilateral knot projections, and a 4N upper bound for the genus-one bracket span.