Knotted Objects Confined to Tubes in the Simple Cubic Lattice

Motivated by biological questions related to DNA packing and the movement of molecules through channels, it is of interest to determine whether a specific knot or link type can be realized in a confined volume. In this talk, we will discuss the size of the smallest lattice tube that can contain certain families of knotted objects. We will take advantage of a theorem of Arsuaga et al., which allows us to study entanglements in lattice tubes by analyzing how level spheres coming from the standard height function intersect the knotted object. We conclude by discussing the exponential growth rate of links in the smallest lattice tube which admits nontrivial knotting and linking. This talk is based on joint work with Jeremy Eng, Robert Scharein, and Chris Soteros.