On the Lower Bounds of Ramsey Numbers of Knots

The Ramsey number is known for only a few specific knots and links, namely the Hopf link and the trefoil knot (although not published in periodicals). We establish the lower bound of all Ramsey numbers of any knot to be one greater than its stick number. 1 Background and Definitions The study of Ramsey numbers of knots can be found at the intersection of knot theory and graph theory. 1.1 Knot Theory Background A knot is a simple closed curve in <3, while a link is a set of disjoint knots. As shown in figure 1 the unknot(a), trefoil knot(b), figure-8 knot(c), unlink(d), and Hopf link(e) are examples of inequivalent links. Figure 1 Some simple knots Stick knots are knots composed of straight line segments intersecting only two at a time. The stick number, s(k), of a knot k, is the fewest number of sticks necessary to embed a knot in <3. Many stick numbers for knots are known (MM). For example, s(unknot) = 3, s(unlink) = 6, s(trefoil) = 6, and s(figure − 8) = 7. Illustrations of these are in Figure 2. Also, Calvo has classified all 8-stick knots(JC).