Maps on Surfaces and the Tabulation of Knots and Links in the Thickened Torus Through Ten Crossings

Abstract

We give a combinatorial tabulation of knots and links in the thickened torus $T^2 \times I$ based on the theory of maps on surfaces: cellular $4$-regular torus projections are encoded by permutation pairs, and unsensed equivalence classes are enumerated completely and without duplication by canonical representatives. The method is validated against published genus-one tables for $N \le 5$ and extended to $N=6,7,8,9,10$, producing, to our knowledge, the first complete tabulation of prime knot and link diagram types in the fixed thickened torus $T^2 \times I$ at these crossing numbers, under the conventions stated below -- more than $1.3$ million knot and link types in total. The computation yields three proved structural results on straight-ahead components, bigon faces, and the parity of the $a$-span of the genus-one bracket, together with several conjectures, including a $4N$ bound on the $a$-span in the spirit of the Kauffman--Murasugi--Thistlethwaite theorem. A reference implementation and machine-readable datasets are provided.