Local Structure of Ideal Shapes of Knots, II, Constant Curvature Case

The thickness, NIR(K) of a knot or link K is defined to be the radius of the largest solid tube one can put around the curve without any self intersections, which is also known as the normal injectivity radius of K. For C^{1,1} curves K, NIR(K)=min{(1/2)DCSC(K),(1/(supkappa(K))))}, where kappa(K) is the generalized curvature, and the double critical self distance DCSD(K) is the shortest length of the segments perpendicular to K at both end points. The knots and links in ideal shapes (or tight knots or links) belong to the minima of ropelength = length/thickness within a fixed isotopy class. In this article, we prove that NIR(K)=(1/2)DCSC(K), for every relative minimum K of ropelength in R^n for certain dimensions n, including n=3.