On the densest packing of polycylinders in any dimension

Using transversality and a dimension reduction argument, a result of A. Bezdek and W. Kuperberg is applied to polycylinders $\mathbb{D}^2\times \mathbb{R}^n$, showing that the optimal packing density is $\pi/\sqrt{12}$ in any dimension.Comment: Edited to reflect acknowledgements in the published versio

A Gordian Pair of Links

We construct a pair of isotopic link configurations that are not thick isotopic while preserving total length.Comment: 2 pages, 1 figur

Bounds on packing density via slicing

This document is composed of a series of articles in discrete geometry, each solving a problem in packing density. • The first proves a local upper bound for the packing density of regular pentagons in R2. By reducing a nonlinear programming problem to a linear one, computational methods show that the conjectured global optimal solution is locally optimal. • The second proves an upper bound for the packing density of finite cylinders in R3. Using a measure theoretic approach to estimate boundary error, the first bound that is asymptotically sharp with respect to the length of the cylinder is found. This gives the first sharp upper bound for the packing density of half-infinite cylinders as a corollary. • The third proves an upper bound for the packing density of infinite polycylinders in Rn. Using transversality and a dimension reduction argument, an existing result for R3 is applied to Rn. This gives the first non-trivial sharp upper bound for the packing density of any object in dimensions four and greater