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The thickness, NIR(K) of a knot or link K is defined to be the radius of thelargest 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} curvesK, NIR(K)=min{(1/2)DCSC(K),(1/(supkappa(K))))}, where kappa(K) is thegeneralized curvature, and the double critical self distance DCSD(K) is theshortest length of the segments perpendicular to K at both end points. Theknots and links in ideal shapes (or tight knots or links) belong to the minimaof 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 ropelengthin R^n for certain dimensions n, including n=3.

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