Ramsey Numbers and Other Knot Invariants

We make use of a particular linear spatial embedding, the cyclic polytope, in an exploration of bounds on the Ramsey number of knots. Using arc presentations to simplify knotted cycles of this embedding, we examine the relationships between the Ramsey number, bridge number, crossing number, stick number and arc index of knots. In particular we show the Ramsey number is at least as large as the sum of the bridge number and the arc index, and at least as large as the sum of the crossing number and the bridge number plus 2. We also show that for a particular class of torus knots, Tp−1,p, the difference between the Ramsey number and stick number grows without bound.