Classification of prime links in the thickened torus having crossing number 5

Abstract

The goal of this paper is to tabulate all prime links in the thickened torus [Formula: see text] having diagrams having crossing number 5. First, we construct a table of prime projections of links on the torus [Formula: see text] having exactly 5 crossings. To this end, we enumerate abstract quadrivalent graphs of special type and consider all possible embeddings of the graphs into the torus [Formula: see text] in order to construct prime projections. Then, we prove that all obtained projections are inequivalent. Second, we use the list of prime projections to construct a table of diagrams of prime links in the torus [Formula: see text]. In order to prove that all those links are inequivalent, we use two modifications of the Kauffman bracket. Several known and new tricks allow us to keep the process within reasonable limits and rigorously theoretically prove the completeness of the constructed tables.

Introduction to Periodic Geometry and Topology

“…Definition 1 (a lattice 𝚲, a unit cell 𝑼, a motif 𝑴, a periodic set 𝑺 = 𝚲 + 𝑴) For a lin- A primitive unit cell 𝑈 of any lattice has a motif of one point counted with weights as follows. Points strictly inside 𝑈 have weight 1, points inside the faces of 𝑈 have weight 1 2 , corners of 𝑈 ⊂ R 2 have weight 1 4 and so on. All unit cells in Fig.…”

“…, when the shortest interval [ 𝑝 𝑖 , 𝑝 𝑖+1 ] of the length 𝑑 [1] is covered by the balls centered at 𝑝 𝑖 , 𝑝 𝑖+1 . At this moment, all 𝑚 intervals cover the subregion of the length 𝑚𝑑 [1] . Then the graph of 𝜓 0 (𝑡) has the first corner points (0, 1) and…”

“…The above pattern generalizes to the 𝑖-th critical radius 𝑡 = 1 2 𝑑 [𝑖 ] , when the covered subregion has the length (for the still growing intervals). For the final critical radius 𝑡 = 1 2 𝑑 [𝑚] , the whole interval…”

“…2 give 𝜓 0 with the corner points (0, 1), If (say) 𝑑 𝑖−1 < 𝑑 𝑖 , then the subinterval covered only by [ 𝑝 𝑖 − 𝑡, 𝑝 𝑖 + 𝑡] is shrinking on the left and is growing at the same rate on the right until it touches the growing interval centered at the right neighbor. During this period, when 𝑡 is between 1 2 𝑑 𝑖−1 and 1 2 𝑑 𝑖 , the function 𝜂 1𝑖 (𝑡) = 𝑑 remains constant. If 𝑑 𝑖−1 = 𝑑 𝑖 , this horizontal piece collapses to one point in the graph of 𝜂 1𝑖 (𝑡).…”

“…So the graph of 𝜂 1𝑖 has a trapezoid form with the corner points (0, 0), ( 𝑑 𝑖−1 2 , 𝑑), ( 𝑑 𝑖 2 , 𝑑), ( 𝑑 𝑖−1 +𝑑 𝑖 2 , 0). In Example 9 for 𝑆 = {0, 1 3 , 1 2 } + Z, the distances 𝑑 1 = 1 3 , 𝑑 2 = 1 6 , 𝑑 3 = 1 2 = 𝑑 0 give 𝜂 11 = 𝜂 𝑅 with the corner points (0, 0), The above argument is similar to the proof of (10𝑏), which can be considered as the partial case of (10c) for 𝑘 = 1 if we replace all empty sums by 0. In Example 9 for 𝑆 = {0, 1 3 , 1 2 } + Z, we have…”