A Computational Approach to Knotting in Complete Graphs Dana Rowland and David Toth Merrimack College, North Andover, MA Abstract We are interested in finding an embedding of the complete graph on nine vertices that is minimally knotted. We conjecture that every embedding of a complete graph with nine vertices contains at least one knot that is more complicated than the trefoil knot. Checking all of the embeddings obtained by changing crossings requires significant computing power which is far beyond what is available on our campus. We have been using the TeraGrid to test embeddings and have checked 2 36 embeddings, all of which have contained a non-trivial, non-trefoil knot, which supports our conjecture. This has been accomplished in a fraction of the time it would have taken to do so with the compute power on our campus. We will continue to use the TeraGrid to explore the minimum number of knotted cycles. Introduction Take a piece of string, stretch it, tangle and twist it up, and then attach the ends together. The result is a mathematical knot. Two such knots are considered to be equivalent if one can be deformed and stretched into the same position as the other, without cutting the string. If the “knot” is just an unknotted circle, we say it is the trivial knot. The simplest non-trivial knot is a trefoil knot, shown in Figure 1. Determining when two given knots are equivalent is one of the central problems in knot theory. A graph is a set of vertices, where some of the vertices are connected to each other by edges. A complete graph is a graph where every pair of vertices is connected by an edge. An embedding of a graph is a particular way to connect the vertices in three dimensional space. A Hamiltonian cycle is a closed loop in the graph that passes through each vertex exactly once and returns to the starting vertex. Each Hamiltonian cycle in an embedding gives us a mathematical knot. Figure 2 illustrates such a 3-dimensional graph; the gaps, or crossings, denote where one edge passes “over” another one. Note that changing which edge is on top at a crossing gives a different embedding of the graph, and can change the knot type of a Hamiltonian cycle that contains that crossing. There are 20,160 Hamiltonian cycles in Figure 2. Some of them, such as the roundtrip path which follows the vertices in order (1,2,3,4,5,6,7,8,9,1), can be deformed into an unknotted circle. Other paths, however, cannot be untangled, such as (1,3,8,6,4,2,9,5,7,1) shown above. In fact, work of Conway and Gordon [2] implies that no matter how one initially connects the pairs of points in a complete graph on nine vertices, at least eight of the resulting Hamiltonian cycles are knotted. However, no known example of an embedding realizes this lower bound. Methods For each cycle in an embedding, we first determine the Dowker- Thistlethwaite (DT) code [3]. The DT code is a sequence of even integers that describe the knot, and can be found by using the following method. Pick a Hamiltonian cycle and choose a starting vertex and a direction. Follow the edge from vertex to next vertex, eventually ending at the starting vertex. Every time you reach an intersection of two edges, give it a number. Start the numbering with 1 and increase it by one every time. Since each crossing will be encountered twice this way, the numbers will run from 1 to (2 * the number of crossings) and each crossing will be labeled with an even number and an odd number. For even numbered labels, if the edge you are following crosses under the other edge, make the number negative. Figure 3 illustrates the DT code for a knot. In this particular knot, there are no negative numbers because as we follow the cycle, the even numbers are assigned when we go over the other crossings. List the even numbers matched with 1, 3, 5, … to get the sequence of even numbers that is called the DT code. Results Every embedding obtained by making crossing changes to the embedding in Figure 2 contains at least one non-trivial non-trefoil knotted Hamiltonian cycle! We have identified embeddings that have exactly one non-trivial, non- trefoil knotted cycle—but the knot is somewhat complex: the 8 19 knot. (See Figure 1.) We have identified 4 embeddings that contain only trefoils and figure- eight knots. This disproves a conjecture that every embedding of K 9 contains a cycle that has degree four Conway polynomial. Future Work We will continue to use the TeraGrid to explore the minimum number of knotted cycles in various ways: The minimum number of knotted cycles in an embedding of K 8 is known to be between 3 and 21. We will attempt to improve the upper bound by searching computationally. While it is theoretically possible that an embedding of K 9 contains only unknots and trefoils as cycles, no known example exists. We will extend our search by considering more complicated embeddings of K 9. Embeddings of K 9 must contain at least 24 knotted Hamiltonian cycles. We intend to establish upper bounds for total number of knotted cycles, and number of knotted cycles with restrictions on knot- type (for example, only knots with fewer than 4 crossings). References [1] Scharein, Rob. ( ). KnotPlot: Hypnogogic Software [2] John H. Conway and Cameron Gordon (1983). Knots and links in spatial graphs. J. Graph Theory, 7(4): [3]C. H. Dowker and Morwen B. Thistlethwaite (1983). Classification of knot projections. Topology and its Applications 16 (1): 19–31 Figure 1 – Some distinct knots, including the trivial knot, the trefoil knot, the figure-eight knot, and the 8 19 knot. Pictures created using KnotPlot[1]. Figure 2 – An embedding of K 9 with a knotted cycle highlighted Figure 3 – The cycle shown has DT code 6824