Sphere Number of Simple Hexagonal Lattice

Lattices are collections of points in 3-space generated by integral linear combinations of vectors in 3-space. Lattices naturally occur as the centers of solid polyhedra that tesselate 3-space. The simple hexagonal lattice is the collection of centers of hexagonal prisms that tesselate 3-space, or can be viewed as combinations of the vectors $(1,0,0)$, $(1/2, (\sqrt{3})/2, 0)$, and $(0,0,1)$. One defines a lattice sphere of radius $r$ centered at the origin to be the collection of lattice points that are within a distance $r$ of the origin. Lattice knots are closed non-intersecting polygons whose vertices lie on the lattice. Our project focuses on finding the smallest radius of a lattice sphere containing a non-trivial lattice knot - this number is the sphere number of the lattice.