AVERAGE CROSSING NUMBER, TOTAL CURVATURE AND ROPELENGTH OF THICK KNOTS

Abstract

Let K be a smooth knot of unit thickness embedded in the space [Formula: see text] with length L(K) and total curvature κ(K). Then [Formula: see text] where acn (K) is the average crossing number of the embedded knot K and c > 0 is a constant independent of the knot K. This relationship had been conjectured in [G. Buck and J. Simon, Total curvature and packing of knots, Topology Appl.154 (2007) 192–204] where it is shown that the square root power on the curvature is the lowest possible. In the last section we give several examples to illustrate some relationships between the three quantities average crossing number, total curvature and ropelength.