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.

翻译：我们提出了一种基于曲面地图理论，用于在加厚环面T^2 x I上穷举生成和制表纽结与链环图的算法框架。通过置换对(α, σ)编码环面上的4-正则胞腔投影，并利用典范代表元完整无重复地枚举无向等价类。交叉分配、局部图级约简以及广义考夫曼型括号均在相同的置换模型中表述。该流程与已发表的亏格一分类结果（交叉数N ≤ 5）进行验证，随后扩展至N = 6、7、8，据我们所知，在所述比较约定下首次完整给出了这些交叉数下的亏格一列表。最终数据集包含超过33,000种纽结与链环类型。除表格外，计算还证明了若干结构性事实，包括关于考夫曼括号a-跨度的奇偶性陈述，以及4-正则环面地图中双角面数量的严格上界N-1。此外，计算引出了多个猜想，其中包括直行分支最大数量的公式、等四边形纽结投影的不存在性，以及亏格一括号跨度的4N上界。