The embedding space of hexagonal knots

The topology of the space of rooted oriented hexagonal knots embedded in R3 is described, with special attention given to the number of components that make up this space and to the topological knot types which they represent. Two cases are considered: (i) hexagons with varying edge length, and (ii) equilateral hexagons with unit-length edges. The structure of these spaces then gives new notions of “hexagonal knottedness”. In each case, the space consists of five components, but contains only three topological knot types. Therefore each type of “hexagonal equivalence” is strictly stronger than topological equivalence. In particular, unlike their topological counterparts, hexagonal trefoils are not reversible; thus there are two distinct components containing each type of topological trefoil. The inclusion of equilateral hexagons into the larger class of hexagons with arbitrary edge length maps hexagonal knot types bijectively; however the kernel of this inclusion at the level of fundamental group is shown to be non-trivial. In addition, combinatorial invariants are developed to distinguish between the five different “hexagonal knot types”, giving a complete classification of hexagonal knots.