Computable invariants for links in the three-torus

Abstract

This work contributes to our knowledge of a type of crystal called rod packings by viewing them as links in the three-torus and computing the fundamental group of their complementary space, which is a topological invariant for these structures. Topological notions play an important role in chemistry. Chemists need the topological properties of crystalline materials for better understanding of their connectivity and classification [Ockwig et al., 2005]. We extend some standard invariants in knot theory from the normal case of knots and links in the three-torusto the case of knots and links in the three-torus. The fundamental group of the complement of knots and links is selected in this study. Although this invariant is not a complete invariant as we show in Chapter 3, it is still a strong invariant to compute. It is possible to compute this invariant by hand for simple examples in this thesis, and there are practical algorithms for computing the presentation of more complicated examples such as helical weavings [Evans et al., 2013]. We compute this invariant to classify the structures described in O'Keeffe et al. [2001] topologically, then we compare our equivalency classes with ambient isotopy of weavings [Evans et al., 2013] produced by a physical simulation for these periodic structures (called PB-SONO algorithm). In Grishanov et al. [2009], doubly periodic structures are considered for computing some numerical invariants, which leads us to enumerate some easier invariants such as the number of components. However, these invariants are not sufficient to discriminate the 3-periodic structures in our case, since some of them have the same invariants. For our computations, we use the standard techniques in knot theory from Rolfsen [1976] and some works on the topology of the three-torus from Johnson [2006]. As it was mentioned, we compute the fundamental group of the complement of these links (structures from crystallography). This invariant is sufficient to prove these structures are distinct within the 3-torus. Then we generalise the example of Whitehead links [Whitehead, 1937] in the 3-torus by using a twisting homeomorphism in a handlebody (with boundary of P-surface) from the standard Heegaard splitting of the 3-torus. This shows that the complementary space is not a complete invariant for links in the 3-torus. Show more