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We consider the expected value for the total curvature of a random closed polygon. Numerical experiments have suggested that as the number of edges becomes large, the difference between the expected total curvature of a random closed polygon and a random open polygon with the same number of turning angles approaches a positive constant. We show that this is true for a natural class of probability measures on polygons, and give a formula for the constant in terms of the moments of the edge-length distribution. We then consider the symmetric measure on closed polygons of fixed total length constructed by Cantarella, Deguchi, and Shonkwiler. For this measure, we are able to prove that the expected value of total curvature for a closed n-gon is exactly π/2 n + π/4 2n/2n-3 . As a consequence, we show that at least 1/3 of fixed-length hexagons and 1/11 of fixed-length heptagons in R3 are unknotted. [ABSTRACT FROM AUTHOR]

We present new computations of tight shapes obtained using the constrained gradient descent code ridgerunner for 544 composite knots with 12 and fewer crossings, expanding our dataset to 943 knots and links. We use the new data set to analyze two outstanding conjectures about tight knots, namely that the ropelengths of composite knots are at least 4π - 4 less than the sums of the prime factors and that the writhes of composite knots are the sums of the writhes of the prime factors. Our numerics support the connect sum conjecture and argue against the additivity of writhe conjecture. We also present data on the number of configurations having straight segments and highly curved kinked regions. [ABSTRACT FROM AUTHOR]

We find bounds on the difference between the writhing numbers of a smooth curve and a polygonal curve inscribed within. The proof is based on an extension of Fuller's difference of writhe formula to the case of polygonal curves. The results establish error bounds useful in the numerical computation of writhe in terms of bounds on the edge lengths of the polygon and the derivatives of the curve. The bounds are "adaptive" in the sense that they improve when regions of the smooth curve with larger derivatives are approximated by shorter edges of the polygon. [ABSTRACT FROM AUTHOR]

This paper provides bounds for the ropelength of a link in terms of the crossing numbers of its prime components. As in earlier papers, the bounds grow with the square of the crossing number; however, the constant involved is a substantial improvement on previous results. The proof depends essentially on writing links in terms of their arc-presentations, and has as a key ingredient Bae and Park''s theorem that an n-crossing link has an arc-presentation with less than or equal to n+2 arcs. [Copyright &y& Elsevier]